   Chapter 5.5, Problem 25E ### 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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# Finding the Area Bounded by Two Graphs In Exercises 15-30, sketch the region bounded by the graphs of the functions and find the area of the region. See Examples 1, 2, 3, and 4. y = x e − x 2 , y = − 1 , x = 0 , x = 1

To determine

To graph: The region bounded by the graphs of y=xex2, y=1, x=0 and x=1, and also compute the area of the region.

Explanation

Given Information:

The region bounded by the graphs of y=xex2, y=1, x=0 and x=1.

Graph:

Consider the provided functions,

y=xex2

And, y=1

Consider the first function y=xex2

It is an exponential function with negative exponent. The graph of the function will lie in first and third quadrant.

Compute the x-intercepts of the function,

xex2=0x=0or  ex2=0   (not possible)

The x-intercept is x=0.

It will also be the y-intercept.

Consider the second function y=1

It is the equation of a horizontal line with y-intercept y=1

Sketch the graph of two functions as follows:

Formula used:

The derivative of d(xn)dx=nxn1.

The integration of xndx=xn+1n+1+c and exdx=ex+c.

Area of a region bounded by two graphs is calculated using the following formula,

If f and g are continuous on [a,b] and g(x)f(x) for all x in [a,b], then the area of the region bounded by the graphs of f, g, x=a and x=b is given by

A=ab[f(x)g(x)]dx

Calculation:

From the graph, 1xex2 for all x in the interval [0,1]. Assume f(x)=xex2 and g(x)=1, use the formula for the area to calculate the area of the provided region,

Area=01[f(x)g(x)]dx=01[xex20]dx=01xex2dx

Assume that u=x2, then dudx=2x

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