   Chapter 6.1, Problem 48E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Area of a Region In Exercises 47-52, find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. y = ( x 2 − 1 ) e x , y = 0 , x = − 1 , x = 1

To determine

To calculate: The area of the region bounded by graphs of the equations y=(x21)ex,y=0,x=1,x=1 and then verify the result by using graphing utility.

Explanation

Given Information:

The provided equations are y=(x21)ex,y=0,x=1,x=1.

Formula used:

Integration by parts.

When u and v is assumed to be the differentiable functions of x then,

u dv=uvv du

Calculation:

Consider the equation,

y=(x21)ex.

Here the lower limit is x=1 and the upper limit is x=1. So, the area of the bounded regions would be,

Area=11(x21)exdx=11(1x2)exdx

Let u=1x2 and dv=exdx, then

dv=exdx

Apply integral on both sides of the above equation as,

dv=exdxv=ex

Thus, v=ex,

Then differentiate both sides of the equation u=1x2 as,

du=d(1x2)du=2x dx

So, du=2x dx

Now apply integration by parts formula.

Substitute u=1x2, dv=exdx, v=ex and du=2x dx in the formula u dv=uvv du as,

(1x2)ex dx=(1x2)ex+2x(ex)

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