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Calculus

James Stewart

Chapter 4

Integrals - all with Video Answers

Educators

Section 1

4.1 Exercises

04:01

Problem 1

(a) By reading values from the given graph of $f$, use four rectangles to find a lower estimate and an upper estimate for the area under the given graph of $f$ from $x=0$ to $x=8$. In each case sketch the rectangles that you use.
(b) Find new estimates using eight rectangles in each case.
(GRAPH CAN'T COPY)

Lucas Finney
Lucas Finney
Numerade Educator
17:14

Problem 2

(a) Use six rectangles to find estimates of each type for the area under the given graph of $f$ from $x=0$ to $x=12$.
(i) $L_6 \quad$ (sample points are left endpoints)
(ii) $R_6 \quad$ (sample points are right endpoints)
(iii) $M_6$ (sample points are midpoints)
(b) Is $L_6$ an underestimate or overestimate of the true area?
(c) Is $R_6$ an underestimate or overestimate of the true area?
(d) Which of the numbers $L_6, R_6$, or $M_6$ gives the best estimate? Explain.

EW
Eleanor Wang
Numerade Educator
01:16

Problem 3

(a) Estimate the area under the graph of $f(x)=\cos x$ from $x=0$ to $x=\pi / 2$ using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
09:48

Problem 4

(a) Estimate the area under the graph of $f(x)=\sqrt{x}$ from $x=0$ to $x=4$ using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints.

Ashka Dalal
Ashka Dalal
Numerade Educator
09:00

Problem 5

(a) Estimate the area under the graph of $f(x)=1+x^2$ from $x=-1$ to $x=2$ using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.
(b) Repeat part (a) using left endpoints.
(c) Repeat part (a) using midpoints.
(d) From your sketches in parts (a)-(c), which appears to be the best estimate?

Frank Lin
Frank Lin
Numerade Educator
15:48

Problem 6

(a) Graph the function
$$f(x)=1 /\left(1+x^2\right) \quad-2 \leqslant x \leqslant 2$$
(b) Estimate the area under the graph of $f$ using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles.
(c) Improve your estimates in part (b) by using eight rectangles.

Mukesh Tumbapo
Mukesh Tumbapo
Numerade Educator
18:57

Problem 7

Evaluate the upper and lower sums for $f(x)=2+\sin x$, $0 \leqslant x \leqslant \pi$, with $n=2,4$, and 8 . Illustrate with diagrams like Figure 14.

Ashka Dalal
Ashka Dalal
Numerade Educator
View

Problem 8

Evaluate the upper and lower sums for $f(x)=1+x^2$, $-1 \leqslant x \leqslant 1$, with $n=3$ and 4 . Illustrate with diagrams like Figure 14.

Thomas Tamanaha
Thomas Tamanaha
Numerade Educator
01:05

Problem 9

9-10 With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $n$, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $n=10,30,50$, and 100 . Then guess the value of the exact area.
The region under $y=x^4$ from 0 to 1

James Kiss
James Kiss
Numerade Educator
01:14

Problem 10

With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $n$, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $n=10,30,50$, and 100 . Then guess the value of the exact area.
The region under $y=\cos x$ from 0 to $\pi / 2$

James Kiss
James Kiss
Numerade Educator
01:54

Problem 11

Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if $x^*$ is a left or right endpoint. (For instance, in Maple use leftbox, rightbox, leftsum, and rightsum.)
(a) If $f(x)=1 /\left(x^2+1\right), 0 \leqslant x \leqslant 1$, find the left and right sums for $n=10,30$, and 50 .
(b) Illustrate by graphing the rectangles in part (a).
(c) Show that the exact area under $f$ lies between 0.780 and 0.791 .

James Kiss
James Kiss
Numerade Educator
02:25

Problem 12

(a) If $f(x)=x /(x+2), 1 \leqslant x \leqslant 4$, use the commands discussed in Exercise 11 to find the left and right sums for $n=10,30$, and 50 .
(b) Illustrate by graphing the rectangles in part (a).
(c) Show that the exact area under $f$ lies between 1.603 and 1.624 .

Sam Low
Sam Low
Numerade Educator
02:27

Problem 13

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.
(TABLE CAN'T COPY)

Frank Lin
Frank Lin
Numerade Educator
06:04

Problem 14

Speedometer readings for a motorcycle at 12 -second intervals are given in the table.
(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals.
(b) Give another estimate using the velocities at the end of the time periods.
(c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.
(TABLE CAN'T COPY)

KW
Kaitlyn Walker
Numerade Educator
03:03

Problem 15

Oil leaked from a tank at a rate of $r(t)$ liters per hour. The rate decreased as time passed and values of the rate at twohour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.
(TABLE CAN'T COPY)

Carolyn Behr-Jerome
Carolyn Behr-Jerome
Numerade Educator
01:08

Problem 16

When we estimate distances from velocity data, it is sometimes necessary to use times $t_0, t_1, t_2, t_3, \ldots$ that are not equally spaced. We can still estimate distances using the time periods $\Delta t_i=t_i-t_{i-1}$. For example, on May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height above the earth's surface of the Endeavour, 62 seconds after liftoff.
(TABLE CAN'T COPY)

Frank Lin
Frank Lin
Numerade Educator
03:19

Problem 17

The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.
(GRAPH CAN'T COPY)

Lucas Finney
Lucas Finney
Numerade Educator
03:06

Problem 18

The velocity graph of a car accelerating from rest to a speed of $120 \mathrm{~km} / \mathrm{h}$ over a period of 30 seconds is shown. Estimate the distance traveled during this period.
(GRAPH CAN'T COPY)

Lucas Finney
Lucas Finney
Numerade Educator
05:08

Problem 19

19-21 Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.
$f(x)=\frac{2 x}{x^2+1}, \quad 1 \leqslant x \leqslant 3$

Brocha Siff
Brocha Siff
Numerade Educator
02:49

Problem 20

Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.
$f(x)=x^2+\sqrt{1+2 x}, \quad 4 \leqslant x \leqslant 7$

Sam Low
Sam Low
Numerade Educator
02:01

Problem 21

Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.
$f(x)=\sqrt{\sin x}, \quad 0 \leqslant x \leqslant \pi$

Sam Low
Sam Low
Numerade Educator
02:31

Problem 22

22-23 Determine a region whose area is equal to the given limit. Do not evaluate the limit.
$\lim _{n \rightarrow \infty} \sum_{i=1}^n \frac{2}{n}\left(5+\frac{2 i}{n}\right)^{10}$

Lucas Finney
Lucas Finney
Numerade Educator
02:19

Problem 23

Determine a region whose area is equal to the given limit. Do not evaluate the limit.
$\lim _{n \rightarrow \infty} \sum_{i=1}^n \frac{\pi}{4 n} \tan \frac{i \pi}{4 n}$

Carson Merrill
Carson Merrill
Numerade Educator
01:51

Problem 24

(a) Use Definition 2 to find an expression for the area under the curve $y=x^3$ from 0 to 1 as a limit.
(b) The following formula for the sum of the cubes of the first $n$ integers is proved in Appendix E. Use it to evaluate the limit in part (a).
$$1^3+2^3+3^3+\cdots+n^3=\left[\frac{n(n+1)}{2}\right]^2$$

Frank Lin
Frank Lin
Numerade Educator
02:38

Problem 25

Let $A$ be the area under the graph of an increasing continuous function $f$ from $a$ to $b$, and let $L_n$ and $R_n$ be the approximations to $A$ with $n$ subintervals using left and right endpoints, respectively.
(a) How are $A, L_{\mathrm{n}}$, and $R_{\mathrm{n}}$ related?
(b) Show that
$$R_n-L_n=\frac{b-a}{n}[f(b)-f(a)]$$
Then draw a diagram to illustrate this equation by showing that the $n$ rectangles representing $R_s-L_s$ can be reassembled to form a single rectangle whose area is the right side of the equation.
(c) Deduce that
$$R_n-A<\frac{b-a}{n}[f(b)-f(a)]$$

James Kiss
James Kiss
Numerade Educator
03:03

Problem 26

If $A$ is the area under the curve $y=\sin x$ from 0 to $\pi / 2$, use Exercise 25 to find a value of $n$ such that $R_n-A<0.0001$.

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
View

Problem 27

(a) Express the area under the curve $y=x^5$ from 0 to 2 as a limit.
(b) Use a computer algebra system to find the sum in your expression from part (a).
(c) Evaluate the limit in part (a).

Gio Maya
Gio Maya
Numerade Educator
04:28

Problem 28

(a) Express the area under the curve $y=x^4+5 x^2+x$ from 2 to 7 as a limit.
(b) Use a computer algebra system to evaluate the sum in part (a).
(c) Use a computer algebra system to find the exact area by evaluating the limit of the expression in part (b).

Sam Low
Sam Low
Numerade Educator
05:43

Problem 29

Find the exact area under the cosine curve $y=\cos x$ from $x=0$ to $x=b$, where $0 \leqslant b \leqslant \pi / 2$. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if $b=\pi / 2$ ?

Robert Daugherty
Robert Daugherty
Numerade Educator
03:20

Problem 30

(a) Let $A_n$ be the area of a polygon with $n$ equal sides inscribed in a circle with radius $r$. By dividing the polygon into $n$ congruent triangles with central angle $2 \pi / n$, show that
$$A_n=\frac{1}{2} n r^2 \sin \left(\frac{2 \pi}{n}\right)$$
(b) Show that $\lim _{n \rightarrow \infty} A_n=\pi r^2$.

Lucas Finney
Lucas Finney
Numerade Educator