   Chapter 13.2, Problem 36E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# For which of the following functions f(x) does ∫ 0 2 f ( x ) d x give the area between the graph of f(x) and the x-axis from x = 0  to  x = 2 ? (a) f ( x ) = x 2 + 1 (b) f ( x ) = − x 2 (c) f ( x ) = x − 1

To determine

For which function f(x) does 02f(x)dx give the area between the graph f(x) and the x –axis from x=0 to x=2, from the options below.

(a) f(x)=x2+1

(b) f(x)=x2

(c) f(x)=x1

Explanation

Given Information:

The provided options are:

(a) f(x)=x2+1

(b) f(x)=x2

(c) f(x)=x1

Explanation:

Consider the function (a) f(x)=x2+1.

Recall that the area between f(x) and the x –axis on the interval [a,b] is given by abf(x)dx if f(x)0.

Since, the function f(x)=x2+1 is positive in the interval [0,2].

Thus, f(x)0 on the interval [0,2].

Thus, for the function f(x)=x2+1 the integral 02f(x)dx gives the area between the graph f(x) and the x –axis from x=0 to x=2, option (a) is correct.

Consider the function (b) f(x)=x2.

Recall that the area between f(x) and the x –axis on the interval [a,b] is given by abf(x)dx if f(x)0.

Substitute 1 for x in f(x)=x2.

f(1)=12=1

Since the function f(x)=x2 is negative at x=1 in the interval [0,2].

Thus, f(x)0 on the interval [0,2]

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