   Chapter 13.7, 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

# Find the area below the graph of y  =  f ( x ) and above the x-axis for f ( x )   = x 2 e − x and x ≥ 0. Use the graph of the integral of f ( x ) over this interval to find the area.

To determine

To calculate: The area below the graph of y=f(x) and above the x axis for function f(x)=x2ex and find the interval for which f(x)>0 and graph the area for the integral of f(x).

Explanation

Given Information:

The provided function is, f(x)=x2ex.

Formula used:

The area under the curve f(x) between x=a and x=b is given by,

A=abf(x)dx

The simple power rule for the differentiation is,

ddx(xn)=nxn1

The formula for integration by parts is given by:

udv=uvvdu

Calculation:

Consider the provided function, f(x)=x2ex.

Now, use the formula of integration by parts,

udv=uvvdu

To obtain the integral as,

0x2exdx=x2ex|0+02xexdx

Split the second integrand into two parts and let,

u=2x

And,

dv=exdx

Then,

du=2dx

And,

v=exdx=ex

Use the formula of integration by parts again as,

0x2exdx=x2ex|0+02xexdx=limb

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