   Chapter 13.2, Problem 43E ### 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

# Find the area between the curve y = x e x 2 and the x-axis from x = 1 to x = 3.

To determine

To calculate: The area between the curve y=xex2 and the x-axis from x=1 to x=3.

Explanation

Given Information:

The provided curve is y=xex2 and the range of x- axis is from x=1 to x=3.

Formula used:

The area under a curve from a to b is the absolute value of the definite integral of the function calculated with the lower limit a and upper limit b.

To calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

According to the power rule of derivatives:

ddx(xn)=nxn1

If u is a function of x, then

euudx=eu+c

Where c is constant.

Calculation:

The provided curve is y=xex2 and the range of x- axis is from x=1 to x=3.

Thus, the upper limit for x is 3 and the lower limit is 1.

Recall that area under a curve from a to b is the absolute value of the definite integral of the function calculated with the lower limit a and upper limit b.

Thus, the area between the curve and the x-axis is given by the integral 13xex2dx.

Consider the expression (x2).

Differentiate with respect to x.

d(x2)dx=2xd(x2)=2xdx

Thus, d(x2)=2xdx.

Simplify the integral using d(x2)=2xdx

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