   Chapter 6.2, Problem 50E ### 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 In  x 2 , y = 0 , x = 4

To determine

To calculate: The exact area of region bounded by the graphs of equations y=xlnx2, y=0, and x=4.

Explanation

Given Information:

The provided equations are y=xlnx2, y=0, and x=4.

Formula used:

The formula for the integral lnudu is;

lnudu=u(1+lnu)+C

General power differentiation Rule:

ddx[un]=nun1dudx

Calculation:

Consider u=x2.

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

du=2xdx

Consider the provided expression:

y=xlnx2

To find the exact area of region bounded by the graphs of equations y=xlnx2, y=0, and x=4 integrate equation y=xlnx2 with lower limit x=1 and higher limit x=4, which can be mathematically written as:

14ydx=14xlnx2dx

Multiply and divide by 2.

14ydx=12142xlnx2dx=1214(lnx2)2xdx

Here, u=x2 and du=2xdx.

Substitute u for x2, and du for 2xdx.

14ydx=1214lnudu

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

14ydx=12[u(1+lnu)]14

Substitute x2 for u

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