   Chapter 8.5, Problem 56E ### 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 of a Region In Exercises 53-56, sketch the region bounded by the graphs of the functions and find the area of the region. y = sec 2 x 4 , y = 4 − x 2 , x = − 1 , x = 1

To determine

To graph: The area bounded by the graph of y=sec2x4 and y=4x2 for x=1 to x=1.

Explanation

Given Information:

The provided functions are y=sec2x4 and y=4x2 for x=1 to x=1.

Formula used:

Write the formula of definite integral of absec2udu.

absec2nudu=[tannun]ab

Here, n is the any number.

Write the formula of definite integral of absinudu.

abundu=[un+1n+1]ab

Write the formula of definite integral of abdu.

abdu=[u]ab

Graph:

Draw area bounded by graph of y=sec2x4 and y=4x2 for x=1 to x=1.

Consider provided function.

y=sec2x4

At x=1,

yx=1=sec2(14)=1.065

At x=0,

yx=0=sec2(04)=sec20=1

At x=1,

yx=1=sec2(14)=1.065

Now, the table for ordered pair (x,y) for the function y=sec2x4 is shown below:

xy11.0650111.065

Again, consider provided function.

y=4x2

At x=1,

yx=1=4(1)2=3

At x=0,

yx=0=4(0)2=4

At x=1,

yx=1=4(1)2=3

Now, the table for ordered pair (x,y) for the function y=4x2 is shown below:

xy130413

Thus, the resulting graph of y=sec2x4 and y=4x2 for x=1 to x=1 is,

Calculation:

From above graph, the area bounded by the graph of y=sec2x4 and y=4x2 for x=1 to x=1 can be calculated as,

11[(4x2)(sec2x4)]dx=201[(4x2)(sec2x4)]dx

Rewrite

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