   Chapter 6.2, Problem 49E ### 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 45-50, use the integration table in Appendix C to find the exact area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. y = x 2 x 2 + 4 , y = 0 , x = 5

To determine

To calculate: The exact area of region bounded by the graphs of equations y=x2x2+4, y=0, and x=5.

Explanation

Given Information:

The provided equations are y=x2x2+4, y=0, and x=5.

Formula used:

The formula for the integral u2u2+a2du is,

u2u2±a2du=18[u(2u2±a2)u2±a2a4ln|u+u2±a2|]+C

The General power differentiation Rule;

ddx[un]=nun1dudx

Calculation:

Consider u=x.

Differentiate the considered function with respect to x using power rule of differentiation.

du=dx

Consider the provided expression:

y=x2x2+4

To find the exact area of region bounded by the graphs of equations y=x2x2+4, y=0, and x=5, integrate equation y=x2x2+4 with lower limit x=0 and higher limit x=5, which can be mathematically written as:

05ydx=05x2x2+4dx

Rewrite.

05ydx=05x2x2+22dx

Here, a=2, u=x and du=dx.

Substitute u for x, a for 2, and du for dx.

05ydx=05u2u2+a2dx

Use the above formula of integration and solve the above integral as:

05ydx=18[u(2u2+a2)u2+a2a4ln|u+u2+a2|]05

Substitute x for u, and 2 for a

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