Calculus: Early Transcendentals

William Briggs, Lyle Cochran, Bernard Gillet

Chapter 5

Integration - all with Video Answers

Educators

+ 1 more educators

Section 1

Approximating Areas under Curves

Problem 1

Suppose an object moves along a line at $15 \mathrm{m} / \mathrm{s},$ for $0 \leq t<2$ and at $25 \mathrm{m} / \mathrm{s},$ for $2 \leq t \leq 5,$ where $t$ is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for $0 \leq t \leq 5$.

Jacob Steele

Jacob Steele

Numerade Educator

Problem 2

Given the graph of the positive velocity of an object moving along a line, what is the geometrical representation of its displacement over a time interval $[a, b] ?$

Mukesh Devi

Numerade Educator

Problem 3

The velocity in $\mathrm{ft} / \mathrm{s}$ of an object moving along a line is given by $v=f(t)$ on the interval $0 \leq t \leq 8$ (see figure), where $t$ is measured in seconds.
a. Divide the interval [0,8] into $n=2$ subintervals, [0,4] and $[4,8] .$ On each subinterval, assume the object moves at a constant velocity equal to the value of $v$ evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,8] (see part (a) of the figure).
b. Repeat part (a) for $n=4$ subintervals (see part (b) of the figure).
(Check your book to see figure)

Jacob Steele

Numerade Educator

Problem 4

The velocity in $\mathrm{ft} / \mathrm{s}$ of an object moving along a line is given by $v=f(t)$ on the interval $0 \leq t \leq 6$ (see figure), where $t$ is measured in seconds.
a. Divide the interval [0,6] into $n=3$ subintervals, [0,2],[2,4] and $[4,6] .$ On each subinterval, assume the object moves at a constant velocity equal to the value of $v$ evaluated at the left endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure).
b. Repeat part (a) for $n=6$ subintervals (see part (b) of the figure).
(Check your book to see graph)

Gregory Higby

Numerade Educator

Problem 5

The velocity in $\mathrm{ft} / \mathrm{s}$ of an object moving along a line is given by $v=f(t)$ on the interval $0 \leq t \leq 6$ (see figure), where $t$ is measured in seconds.
a. Divide the interval [0,6] into $n=3$ subintervals, [0,2],[2,4] and $[4,6] .$ On each subinterval, assume the object moves at a constant velocity equal to the value of $v$ evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure).
b. Repeat part (a) for $n=6$ subintervals (see part (b) of the figure).
(Check your book to see graph)

Jacob Steele

Numerade Educator

Problem 6

The velocity in $\mathrm{ft} / \mathrm{s}$ of an object moving along a line is given by $v=f(t)$ on the interval $0 \leq t \leq 10$ (see figure), where $t$ is measured in seconds.
a. Divide the interval [0,10] into $n=5$ subintervals. On each subinterval, assume the object moves at a constant velocity equal to the value of $v$ evaluated at the left endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,10]
b. Repeat part (a) using the right endpoints to estimate the displacement on [0,10]
c. Repeat part (a) using the midpoints of each subinterval to estimate the displacement on [0,10]

Gregory Higby

Numerade Educator

Problem 7

Suppose you want to approximate the area of the region bounded by the graph of $f(x)=\cos x$ and the $x$ -axis between $x=0$ and $x=\frac{\pi}{2} .$ Explain a possible strategy.

Jacob Steele

Numerade Educator

Problem 8

Explain how Riemann sum approximations to the area of a region under a curve change as the number of subintervals increases.

Gregory Higby

Numerade Educator

Problem 9

Approximating area from a graph Approximate the area of the region bounded by the graph (see figure) and the $x$ -axis by dividing the interval [1,7] into $n=6$ subintervals. Use a left and right Riemann sum to obtain two different approximations.
(Check your book to see graph)

Jacob Steele

Numerade Educator

Problem 10

Approximating area from a graph Approximate the area of the region bounded by the graph (see figure) and the $x$ -axis by dividing the interval [0,6] into $n=3$ subintervals. Use a left, right, and midpoint Riemann sum to obtain three different approximations.
(Check your book to see graph)

Gregory Higby

Numerade Educator

Problem 11

Suppose the interval [1,3] is partitioned into $n=4$ subintervals. What is the subinterval length $\Delta x$ ? List the grid points $x_{0}, x_{1}, x_{2}$ $x_{3},$ and $x_{4} .$ Which points are used for the left, right, and midpoint Riemann sums?

Willis James

Numerade Educator

Problem 12

Suppose the interval [2,6] is partitioned into $n=4$ subintervals with grid points $x_{0}=2, x_{1}=3, x_{2}=4, x_{3}=5,$ and $x_{4}=6$ Write, but do not evaluate, the left, right, and midpoint Riemann sums for $f(x)=x^{2}$.

Anh Hoang

Numerade Educator

Problem 13

Does a right Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval $[a, b] ?$ Explain.

Jacob Steele

Numerade Educator

Problem 14

Does a left Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and increasing on an interval $[a, b] ?$ Explain.

Gregory Higby

Numerade Educator

Problem 15

Approximating displacement The velocity in $\mathrm{ft} / \mathrm{s}$ of an object moving along a line is given by $v=3 t^{2}+1$ on the interval $0 \leq t \leq 4,$ where $t$ is measured in seconds.
a. Divide the interval [0,4] into $n=4$ subintervals, [0,1] $[1,2],[2,3],$ and $[3,4] .$ On each subinterval, assume the object moves at a constant velocity equal to $v$ evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,4] (see part (a) of the figure).
b. Repeat part (a) for $n=8$ subintervals (see part (b) of the figure).
(Check your book to see figure)

Jacob Steele

Numerade Educator

Problem 16

Approximating displacement The velocity in $\mathrm{ft} / \mathrm{s}$ of an object moving along a line is given by $v=\sqrt{10 t}$ on the interval $1 \leq t \leq 7,$ where $t$ is measured in seconds.
a. Divide the interval [1,7] into $n=3$ subintervals, [1,3] $[3,5],$ and $[5,7] .$ On each subinterval, assume the object moves at a constant velocity equal to $v$ evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [1,7] (see part (a) of the figure).
b. Repeat part (a) for $n=6$ subintervals (see part (b) of the figure).
(Check your book to see figure)

Gregory Higby

Numerade Educator

Problem 17

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into $n$ subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
$$v=2 t+1(\mathrm{m} / \mathrm{s}), \text { for } 0 \leq t \leq 8 ; n=2$$

Jacob Steele

Numerade Educator

Problem 18

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into $n$ subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
$$v=e^{t}(\mathrm{m} / \mathrm{s}), \text { for } 0 \leq t \leq 3 ; n=3$$

Gregory Higby

Numerade Educator

Problem 19

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into $n$ subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
$$v=\frac{1}{2 t+1}(\mathrm{m} / \mathrm{s}), \text { for } 0 \leq t \leq 8 ; n=4$$

Jacob Steele

Numerade Educator

Problem 20

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into $n$ subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
$$v=\frac{t^{2}}{2}+4(\mathrm{ft} / \mathrm{s}), \text { for } 0 \leq t \leq 12 ; n=6$$

Gregory Higby

Numerade Educator

Problem 21

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into $n$ subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
$$v=4 \sqrt{t+1}(\mathrm{mi} / \mathrm{hr}), \text { for } 0 \leq t \leq 15 ; n=5$$

Jacob Steele

Numerade Educator

Problem 22

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into $n$ subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
$$v=\frac{t+3}{6}(\mathrm{m} / \mathrm{s}), \text { for } 0 \leq t \leq 4 ; n=4$$

Gregory Higby

Numerade Educator

Problem 23

Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for $f$ on the given interval and for the given value of $n.$ (Check your book to see figure)
$$f(x)=x+1 \text { on }[1,6] ; n=5$$

Jacob Steele

Numerade Educator

Problem 24

Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for $f$ on the given interval and for the given value of $n.$ (Check your book to see figure)
$$f(x)=\frac{1}{x} \text { on }[1,5] ; n=4$$

Gregory Higby

Numerade Educator

Problem 25

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=x+1 \text { on }[0,4] ; n=4$$

Sheryl Ezze

Numerade Educator

Problem 26

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=9-x \text { on }[3,8] ; n=5$$

Gregory Higby

Numerade Educator

Problem 27

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=\cos x \text { on }\left[0, \frac{\pi}{2}\right] ; n=4$$

Jacob Steele

Numerade Educator

Problem 28

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=\sin ^{-1} \frac{x}{3} \text { on }[0,3] ; n=6$$

Gregory Higby

Numerade Educator

Problem 29

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=x^{2}-1 \text { on }[2,4] ; n=4$$

Jacob Steele

Numerade Educator

Problem 30

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=2 x^{2} \text { on }[1,6] ; n=5$$

Gregory Higby

Numerade Educator

Problem 31

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=e^{x / 2} \text { on }[1,4] ; n=6$$

Jacob Steele

Numerade Educator

Problem 32

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=\ln 4 x \text { on }[1,3] ; n=5$$

Gregory Higby

Numerade Educator

Problem 33

A midpoint Riemann sum Approximate the area of the region bounded by the graph of $f(x)=100-x^{2}$ and the $x$ -axis on [0,10] with $n=5$ subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (check your book to see figure).

Jacob Steele

Numerade Educator

Problem 34

A midpoint Riemann sum Approximate the area of the region bounded by the graph of $f(t)=\cos \frac{t}{2}$ and the $t$ -axis on $[0, \pi]$ with $n=4$ subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (check your book to see figure).

Gregory Higby

Numerade Educator

Problem 35

Free fall On October $14,2012,$ Felix Baumgartner stepped off a balloon capsule at an altitude of almost $39 \mathrm{km}$ above Earth's surface and began his free fall. His velocity in $\mathrm{m} / \mathrm{s}$ during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com)
a. Divide the interval [34,64] into $n=5$ subintervals with the gridpoints $x_{0}=34, x_{1}=40, x_{2}=46, x_{3}=52, x_{4}=58,$ and $x_{5}=64 .$ Use left and right Riemann sums to estimate how far Felix fell while traveling at supersonic speed.
b. It is claimed that the actual distance that Felix fell at supersonic speed was approximately $10,485 \mathrm{m}$. Which estimate in part (a) produced the more accurate estimate?
c. How could you obtain more accurate estimates of the total distance fallen than those found in part (a)?

Jacob Steele

Numerade Educator

Problem 36

Free fall Use geometry and the figure given in Exercise 35 to estimate how far Felix fell in the first 20 seconds of his free fall.

Gregory Higby

Numerade Educator

Problem 37

Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n}$
c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.
d. Calculate the midpoint Riemann sum.
$$f(x)=2 x+1 \text { on }[0,4] ; n=4$$

Jacob Steele

Numerade Educator

Problem 38

Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n}$
c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.
d. Calculate the midpoint Riemann sum.
$$f(x)=2 \cos ^{-1} x \text { on }[0,1] ; n=5$$

Gregory Higby

Numerade Educator

Problem 39

Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n}$
c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.
d. Calculate the midpoint Riemann sum.
$$f(x)=\sqrt{x} \text { on }[1,3] ; n=4$$

Jacob Steele

Numerade Educator

Problem 40

Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n}$
c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.
d. Calculate the midpoint Riemann sum.
$$f(x)=x^{2} \text { on }[0,4] ; n=4$$

Gregory Higby

Numerade Educator

Problem 41

Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n}$
c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.
d. Calculate the midpoint Riemann sum.
$$f(x)=\frac{1}{x} \text { on }[1,6] ; n=5$$

Jacob Steele

Numerade Educator

Problem 42

Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n}$
c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles.
d. Calculate the midpoint Riemann sum.
$$f(x)=4-x \text { on }[-1,4] ; n=5$$

Gregory Higby

Numerade Educator

Problem 43

Riemann sums from tables Evaluate the left and right Riemann sums for $f$ over the given interval for the given value of $n.$
$n=4 ;[0,2]$
$$\begin{array}{|c|c|c|c|c|c|}\hline x & 0 & 0.5 & 1 & 1.5 & 2 \\\hline f(x) & 5 & 3 & 2 & 1 & 1 \\\hline\end{array}$$

Jacob Steele

Numerade Educator

Problem 44

Riemann sums from tables Evaluate the left and right Riemann sums for $f$ over the given interval for the given value of $n.$
$n=8 ;[1,5]$
$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline x & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 & 5 \\\hline f(x) & 0 & 2 & 3 & 2 & 2 & 1 & 0 & 2 & 3 \\\hline\end{array}$$

Gregory Higby

Numerade Educator

Problem 45

Displacement from a table of velocities The velocities (in $\mathrm{mi} / \mathrm{hr}$ ) of an automobile moving along a straight highway over a two-hour period are given in the following table.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline t \text { (hr) } & 0 & 0.25 & 0.5 & 0.75 & 1 & 1.25 & 1.5 & 1.75 & 2 \\\hline v(\mathrm{mi} / \mathrm{hr}) & 50 & 50 & 60 & 60 & 55 & 65 & 50 & 60 & 70 \\\hline\end{array}$$
a. Sketch a smooth curve passing through the data points.
b. Find the midpoint Riemann sum approximation to the displacement on [0,2] with $n=2$ and $n=4.$

Jacob Steele

Numerade Educator

Problem 46

Displacement from a table of velocities The velocities (in $\mathrm{m} / \mathrm{s}$ ) of an automobile moving along a straight freeway over a foursecond period are given in the following table.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline t(s) & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\\hline v(\mathrm{m} / \mathrm{s}) & 20 & 25 & 30 & 35 & 30 & 30 & 35 & 40 & 40 \\\hline\end{array}$$
a. Sketch a smooth curve passing through the data points.
b. Find the midpoint Riemann sum approximation to the displacement on [0,4] with $n=2$ and $n=4$ subintervals.

Gregory Higby

Numerade Educator

Problem 47

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
a. $1+2+3+4+5$
b. $4+5+6+7+8+9$
c. $1^{2}+2^{2}+3^{2}+4^{2}$
d. $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$

Jacob Steele

Numerade Educator

Problem 48

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
a. $1+3+5+7+\cdots+99$
b. $4+9+14+\dots+44$
c. $3+8+13+\cdots+63$
d. $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{49 \cdot 50}$

Anh Hoang

Numerade Educator

Problem 49

Sigma notation Evaluate the following expressions.
a. $\sum_{k=1}^{10} k$
b. $\sum_{k=1}^{6}(2 k+1)$
c. $\sum_{k=1}^{4} k^{2}$
d. $\sum_{n=1}^{5}\left(1+n^{2}\right)$
e. $\sum_{m=1}^{3} \frac{2 m+2}{3}$
f. $\sum_{j=1}^{3}(3 j-4)$
g. $\sum_{p=1}^{5}\left(2 p+p^{2}\right)$
h. $\sum_{n=0}^{4} \sin \frac{n \pi}{2}$

Jacob Steele

Numerade Educator

Problem 50

Evaluating sums Evaluate the following expressions by two methods. (i) Use Theorem $5.1 .$ (ii) Use a calculator.
a. $\sum_{k=1}^{45} k$
b. $\sum_{k=1}^{45}(5 k-1)$
c. $\sum_{k=1}^{75} 2 k^{2}$
d. $\sum_{n=1}^{50}\left(1+n^{2}\right)$
e. $\sum_{m=1}^{75} \frac{2 m+2}{3} \quad$ f. $\sum_{j=1}^{20}(3 j-4)$
g. $\sum_{p=1}^{35}\left(2 p+p^{2}\right)$
h. $\sum_{n=0}^{40}\left(n^{2}+3 n-1\right)$

Anh Hoang

Numerade Educator

Problem 51

Riemann sums for larger values of $n$ Complete the following steps for the given function $f$ and interval.
a. For the given value of $n$, use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator.
b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of $f$ and the $x$ -axis on the interval.
$$f(x)=3 \sqrt{x} \text { on }[0,4] ; n=40$$

Jacob Steele

Numerade Educator

Problem 52

Riemann sums for larger values of $n$ Complete the following steps for the given function $f$ and interval.
a. For the given value of $n$, use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator.
b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of $f$ and the $x$ -axis on the interval.
$$f(x)=x^{2}+1 \text { on }[-1,1] ; n=50$$

Anh Hoang

Numerade Educator

Problem 53

Riemann sums for larger values of $n$ Complete the following steps for the given function $f$ and interval.
a. For the given value of $n$, use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator.
b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of $f$ and the $x$ -axis on the interval.
$$f(x)=x^{2}-1 \text { on }[2,5] ; n=75$$

Jacob Steele

Numerade Educator

Problem 54

Riemann sums for larger values of $n$ Complete the following steps for the given function $f$ and interval.
a. For the given value of $n$, use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator.
b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of $f$ and the $x$ -axis on the interval.
$$f(x)=\cos 2 x \text { on }\left[0, \frac{\pi}{4}\right] ; n=60$$

Anh Hoang

Numerade Educator

Problem 55

Use a calculator and right Riemann sums to approximate the area of the given region.
Present your calculations in a table showing the approximations for $n=10,30,60,$ and 80 subintervals. Make a conjecture about the limit of Riemann sums as $n \rightarrow \infty.$
The region bounded by the graph of $f(x)=12-3 x^{2}$ and the $x$ -axis on the interval [-1,1].

Jacob Steele

Numerade Educator

Problem 56

Use a calculator and right Riemann sums to approximate the area of the given region.
Present your calculations in a table showing the approximations for $n=10,30,60,$ and 80 subintervals. Make a conjecture about the limit of Riemann sums as $n \rightarrow \infty.$
The region bounded by the graph of $f(x)=3 x^{2}+1$ and the $x$ -axis on the interval [-1,1].

Gregory Higby

Numerade Educator

Problem 57

Use a calculator and right Riemann sums to approximate the area of the given region.
Present your calculations in a table showing the approximations for $n=10,30,60,$ and 80 subintervals. Make a conjecture about the limit of Riemann sums as $n \rightarrow \infty.$
The region bounded by the graph of $f(x)=\frac{1-\cos x}{2}$ and the $x$ -axis on the interval $[-\pi, \pi]$.

Jacob Steele

Numerade Educator

Problem 58

Use a calculator and right Riemann sums to approximate the area of the given region.
Present your calculations in a table showing the approximations for $n=10,30,60,$ and 80 subintervals. Make a conjecture about the limit of Riemann sums as $n \rightarrow \infty.$
The region bounded by the graph of $f(x)=\left(2^{x}+2^{-x}\right)$ In 2 and the $x$ -axis on the interval [-2,2].

Gregory Higby

Numerade Educator

Problem 59

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Consider the linear function $f(x)=2 x+5$ and the region bounded by its graph and the $x$ -axis on the interval [3,6] Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.
b. A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the $x$ -axis on an interval $[a, b]$
c. For an increasing or decreasing nonconstant function on an interval $[a, b]$ and a given value of $n,$ the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.

Jacob Steele

Numerade Educator

Problem 60

Riemann sums for a semicircle Let $f(x)=\sqrt{1-x^{2}}.$
a. Show that the graph of $f$ is the upper half of a circle of radius 1 centered at the origin.
b. Estimate the area between the graph of $f$ and the $x$ -axis on the interval [-1,1] using a midpoint Riemann sum with $n=25$.
c. Repeat part (b) using $n=75$ rectangles.
d. What happens to the midpoint Riemann sums on [-1,1] as $n \rightarrow \infty ?$

Anh Hoang

Numerade Educator

Problem 61

Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.
The right Riemann sum for $f(x)=x+1$ on [0,4] with $n=50$.

Jacob Steele

Numerade Educator

Problem 62

Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.
The left Riemann sum for $f(x)=e^{x}$ on $[0, \ln 2]$ with $n=40$

Anh Hoang

Numerade Educator

Problem 63

Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.
The midpoint Riemann sum for $f(x)=x^{3}$ on [3,11] with $n=32$

Jacob Steele

Numerade Educator

Problem 64

Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.
The midpoint Riemann sum for $f(x)=1+\cos \pi x$ on [0,2] with $n=50$

Anh Hoang

Numerade Educator

Problem 65

$\sum_{k=1}^{4} f(1+k) \cdot 1$ is a right Riemann sum for $f$ on the interval
[______, _______] with $n=$ _________.

Jacob Steele

Numerade Educator

Problem 66

$\sum_{k=1}^{4} f(1.5+k) \cdot 1$ is a midpoint Riemann sum for $f$ on the interval
[______, _______] with $n=$ _________.

Jacob Steele

Numerade Educator

Problem 67

$\sum_{k=1}^{4} f(1.5+k) \cdot 1$ is a midpoint Riemann sum for $f$ on the interval
[______, _______] with $n=$ _________.

Jacob Steele

Numerade Educator

Problem 68

$\sum_{k=1}^{8} f\left(1.5+\frac{k}{2}\right) \cdot \frac{1}{2}$ is a left Riemann sum for $f$ on the interval
[______, _______] with $n=$ _________.

Jacob Steele

Numerade Educator

Problem 69

Approximating areas Estimate the area of the region bounded by the graph of $f(x)=x^{2}+2$ and the $x$-axis on [0,2] in the following ways.
a. Divide [0,2] into $n=4$ subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.
b. Divide [0,2] into $n=4$ subintervals and approximate the area of the region using a midpoint Riemann sum. Illustrate the solution geometrically.
c. Divide [0,2] into $n=4$ subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

Jacob Steele

Numerade Educator

Problem 70

Displacement from a velocity graph Consider the velocity function for an object moving along a line.
a. Describe the motion of the object over the interval [0,6]
b. Use geometry to find the displacement of the object between $t=0$ and $t=3$
c. Use geometry to find the displacement of the object between $t=3$ and $t=5$
d. Assuming the velocity remains $30 \mathrm{m} / \mathrm{s}$, for $t \geq 4,$ find the function that gives the displacement between $t=0$ and any time $t \geq 4$
(check your book to see figure).

Gregory Higby

Numerade Educator

Problem 71

Displacement from a velocity graph Consider the velocity function for an object moving along a line.
a. Describe the motion of the object over the interval [0,6].
b. Use geometry to find the displacement of the object between $t=0$ and $t=2$.
c. Use geometry to find the displacement of the object between $t=2$ and $t=5$.
d. Assuming the velocity remains $10 \mathrm{m} / \mathrm{s}$, for $t \geq 5,$ find the function that gives the displacement between $t=0$ and any time $t \geq 5$.
(check your book to see figure).

Jacob Steele

Numerade Educator

Problem 72

Flow rates Suppose a gauge at the outflow of a reservoir measures the flow rate of water in units of $\mathrm{ft}^{3} / \mathrm{hr}$. In Chapter $6,$ we show that the total amount of water that flows out of the reservoir is the area under the flow rate curve. Consider the flow rate function shown in the figure.
a. Find the amount of water (in units of $\mathrm{ft}^{3}$ ) that flows out of the reservoir over the interval [0,4]
b. Find the amount of water that flows out of the reservoir over the interval [8,10]
c. Does more water flow out of the reservoir over the interval [0,4] or [4,6]$?$
d. Show that the units of your answer are consistent with the units of the variables on the axes.

Anh Hoang

Numerade Educator

Problem 73

Mass from density A thin 10 -cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of $\mathrm{g} / \mathrm{cm} .$ In Chapter $6,$ we show that the mass of the rod is the area under the density curve.
a. Find the mass of the left half of the rod $(0 \leq x \leq 5)$
b. Find the mass of the right half of the rod $(5 \leq x \leq 10)$
c. Find the mass of the entire rod $(0 \leq x \leq 10)$
d. Find the point along the rod at which it will balance (called the center of mass).

Jacob Steele

Numerade Educator

Problem 74

Displacement from velocity The following functions describe the velocity of a car (in $\mathrm{mi} / \mathrm{hr}$ ) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval $[0, t]$, where $0 \leq t \leq 3$
(Check your book to see figure)
$$v(t)=\left\{\begin{array}{ll}40 & \text { if } 0 \leq t \leq 1.5 \\50 & \text { if } 1.5<t \leq 3\end{array}\right.$$

Gregory Higby

Numerade Educator

Problem 75

Displacement from velocity The following functions describe the velocity of a car (in $\mathrm{mi} / \mathrm{hr}$ ) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval $[0, t]$, where $0 \leq t \leq 3$
(Check your book to see figure)
$$v(t)=\left\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 2 \\50 & \text { if } 2< t \leq 2.5 \\44 & \text { if } 2.5< t \leq 3\end{array}\right.$$

Jacob Steele

Numerade Educator

Problem 76

Functions with absolute value Use a calculator and the method of your choice to approximate the area of the following regions. Present your calculations in a table, showing approximations using $n=16,32,$ and 64 subintervals. Make a conjecture about the limits of the approximations.
The region bounded by the graph of $f(x)=\left|25-x^{2}\right|$ and the $x$ -axis on the interval [0,10].

Gregory Higby

Numerade Educator

Problem 77

Functions with absolute value Use a calculator and the method of your choice to approximate the area of the following regions. Present your calculations in a table, showing approximations using $n=16,32,$ and 64 subintervals. Make a conjecture about the limits of the approximations.
The region bounded by the graph of $f(x)=\left|1-x^{3}\right|$ and the $x$ -axis on the interval [-1,2].

Jacob Steele

Numerade Educator

Problem 78

Riemann sums for constant functions Let $f(x)=c,$ where $c>0,$ be a constant function on $[a, b] .$ Prove that any Riemann sum for any value of $n$ gives the exact area of the region between the graph of $f$ and the $x$ -axis on $[a, b]$.

Anh Hoang

Numerade Educator

Problem 79

Riemann sums for linear functions Assume the linear function $f(x)=m x+c$ is positive on the interval $[a, b] .$ Prove that the midpoint Riemann sum with any value of $n$ gives the exact area of the region between the graph of $f$ and the $x$ -axis on $[a, b]$

Jacob Steele

Numerade Educator

Problem 80

Shape of the graph for left Riemann sums Suppose a left Riemann sum is used to approximate the area of the region bounded by the graph of a positive function and the $x$ -axis on the interval $[a, b] .$ Fill in the following table to indicate whether the resulting approximation underestimates or overestimates the exact area in the four cases shown. Use a sketch to explain your reasoning in each case.
$$\begin{array}{|l|l|l|}\hline & \text { Increasing on }[a, b] & \text { Decreasing on }[a, b] \\\hline \text { Concave up on }[a, b] & & \\\hline \text { Concave down on }[a, b] & & \\\hline\end{array}$$

Gregory Higby

Numerade Educator

Problem 81

Shape of the graph for right Riemann sums Suppose a right Riemann sum is used to approximate the area of the region bounded by the graph of a positive function and the $x$ -axis on the interval $[a, b] .$ Fill in the following table to indicate whether the resulting approximation underestimates or overestimates the exact area in the four cases shown. Use a sketch to explain your reasoning in each case.
$$\begin{array}{|l|l|l|}\hline & \text { Increasing on }[a, b] & \text { Decreasing on }[a, b] \\\hline \text { Concave up on }[a, b] & & \\\hline \text { Concave down on }[a, b] & & \\\hline\end{array}$$

Jacob Steele

Numerade Educator