   Chapter 13.7, Problem 23E ### 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

# In Problems 23-26, find the area, if it exists, of the region under the graph of y   =   f ( x ) and to the right of x = 1. 23.   f ( x ) = x e x 2

To determine

To calculate: The area of the region under the graph of f(x)=xex2 to the right of x=1 if it exists.

Explanation

Given Information:

The provided function is,

f(x)=xex2

Formula used:

According to the exponential rule of integrals,

exdx=ex+C

If the limit defining the improper integral is a unique finite number, then integral converges else it diverges,

af(x)dx=limbabf(x)dx

Calculation:

Consider the provided function,

f(x)=xex2

Now, the area under its graph to the right of x=1 will be given by,

1xex2dx

Now, use the formula,

af(x)dx=limbabf(x)dx

Multiply and divide it by 2 to rewrite the integral as,

1xex2dx=12limb1b2xex2dx

Now, let x2=t, and then differentiate both the sides,

2xdx=dt

Thus, the integral becomes,

1xex2dx=12limb1b

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