Calculus Early Transcendentals

Howard Anton, Irl Bivens, Stephen Davis

Chapter 5

INTEGRATION - all with Video Answers

Educators

Section 1

An Overview of the Area Problem

Problem 1

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\sqrt{x} ;[a, b]=[0,1]
$$

Joseph L.

Joseph L.

Numerade Educator

Problem 2

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\frac{1}{x+1} ;[a, b]=[0,1]
$$

Joseph L.

Numerade Educator

Problem 3

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\sin x ;[a, b]=[0, \pi]
$$

Joseph L.

Numerade Educator

Problem 4

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\cos x ;[a, b]=[0, \pi / 2]
$$

Joseph L.

Numerade Educator

Problem 5

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\frac{1}{x} ;[a, b]=[1,2]
$$

Joseph L.

Numerade Educator

Problem 6

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\cos x ;[a, b]=[-\pi / 2, \pi / 2]
$$

Joseph L.

Numerade Educator

Problem 7

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\sqrt{1-x^{2}} ;[a, b]=[0,1]
$$

Joseph L.

Numerade Educator

Problem 8

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\sqrt{1-x^{2}} ;[a, b]=[-1,1]
$$

Joseph L.

Numerade Educator

Problem 9

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=e^{x} ;[a, b]=[-1,1]
$$

Joseph L.

Numerade Educator

Problem 10

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\ln x ;[a, b]=[1,2]
$$

Joseph L.

Numerade Educator

Problem 11

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\sin ^{-1} x ;[a, b]=[0,1]
$$

Joseph L.

Numerade Educator

Problem 12

Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.
$$
f(x)=\tan ^{-1} x ;[a, b]=[0,1]
$$

Joseph L.

Numerade Educator

Problem 13

Graph each function over the specified interval. Then use simple area formulas from geometry to find the area function $A(x)$ that gives the area between the graph of the specified function $f$ and the interval $[a, x] .$ Confirm that $A^{\prime}(x)=f(x)$ in every case.
$$
f(x)=3 ;[a, x]=[1, x]
$$

Joseph L.

Numerade Educator

Problem 14

Graph each function over the specified interval. Then use simple area formulas from geometry to find the area function $A(x)$ that gives the area between the graph of the specified function $f$ and the interval $[a, x] .$ Confirm that $A^{\prime}(x)=f(x)$ in every case.
$$
f(x)=5 ;[a, x]=[2, x]
$$

Joseph L.

Numerade Educator

Problem 15

Graph each function over the specified interval. Then use simple area formulas from geometry to find the area function $A(x)$ that gives the area between the graph of the specified function $f$ and the interval $[a, x] .$ Confirm that $A^{\prime}(x)=f(x)$ in every case.
$$
f(x)=2 x+2 ;[a, x]=[0, x]
$$

Joseph L.

Numerade Educator

Problem 16

Graph each function over the specified interval. Then use simple area formulas from geometry to find the area function $A(x)$ that gives the area between the graph of the specified function $f$ and the interval $[a, x] .$ Confirm that $A^{\prime}(x)=f(x)$ in every case.
$$
f(x)=3 x-3 ;[a, x]=[1, x]
$$

Joseph L.

Numerade Educator

Problem 17

Graph each function over the specified interval. Then use simple area formulas from geometry to find the area function $A(x)$ that gives the area between the graph of the specified function $f$ and the interval $[a, x] .$ Confirm that $A^{\prime}(x)=f(x)$ in every case.
$$
f(x)=2 x+2 ;[a, x]=[1, x]
$$

Joseph L.

Numerade Educator

Problem 18

Graph each function over the specified interval. Then use simple area formulas from geometry to find the area function $A(x)$ that gives the area between the graph of the specified function $f$ and the interval $[a, x] .$ Confirm that $A^{\prime}(x)=f(x)$ in every case.
$$
f(x)=3 x-3 ;[a, x]=[2, x]
$$

Joseph L.

Numerade Educator

Problem 19

Determine whether the statement is true or false. Explain your answer.
If $A(n)$ denotes the area of a regular $n$ -sided polygon inscribed in a circle of radius $2,$ then $\lim _{n \rightarrow+\infty} A(n)=2 \pi$.

Joseph L.

Numerade Educator

Problem 20

Determine whether the statement is true or false. Explain your answer.
If the area under the curve $y=x^{2}$ over an interval is approximated by the total area of a collection of rectangles, the approximation will be too large.

Joseph L.

Numerade Educator

Problem 21

Determine whether the statement is true or false. Explain your answer.
If $A(x)$ is the area under the graph of a nonnegative continuous function $f$ over an interval $[a, x],$ then $A^{\prime}(x)=f(x)$

Joseph L.

Numerade Educator

Problem 22

Determine whether the statement is true or false. Explain your answer.
If $A(x)$ is the area under the graph of a nonnegative continuous function $f$ over an interval $[a, x],$ then $A(x)$ will be a
continuous function.

Norman A.

Numerade Educator

Problem 23

Explain how to use the formula for $A(x)$ found in the solution to Example 2 to determine the area between the graph of $y=x^{2}$ and the interval $[3,6]$

Joseph L.

Numerade Educator

Problem 24

Repeat Exercise 23 for the interval $[-3,9]$

Joseph L.

Numerade Educator

Problem 25

Let $A$ denote the area between the graph of $f(x)=\sqrt{x}$
and the interval $[0,1],$ and let $B$ denote the area between the graph of $f(x)=x^{2}$ and the interval $[0,1] .$ Explain geometrically why $A+B=1$

Joseph L.

Numerade Educator

Problem 26

Let $A$ denote the area between the graph of $f(x)=1 / x$ and the interval $[1,2],$ and let $B$ denote the area between the graph of $f$ and the interval $\left[\frac{1}{2}, 1\right] .$ Explain
geometrically why $A=B$.

Joseph L.

Numerade Educator

Problem 27

The area $A(x)$ under the graph of $f$ and over the interval $[a, x]$ is given. Find the function $f$ and the value of $a .$
$$
A(x)=x^{2}-4
$$

Norman A.

Numerade Educator

Problem 28

The area $A(x)$ under the graph of $f$ and over the interval $[a, x]$ is given. Find the function $f$ and the value of $a .$
$$
A(x)=x^{2}-x
$$

Norman A.

Numerade Educator

Problem 29

Writing Compare and contrast the rectangle method and the antiderivative method.

Norman A.

Numerade Educator

Problem 30

Writing Suppose that $f$ is a nonnegative continuous function on an interval $[a, b]$ and that $g(x)=f(x)+C,$ where $C$ is a positive constant. What will be the area of the region between the graphs of $f$ and $g ?$

Norman A.

Numerade Educator