   Chapter 6.1, 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 47-52, find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your results. y = x − 3 ln x , y = 0 , x = e

To determine

To calculate: The area of the region bounded by graphs of the equations y=x3lnx,y=0,x=e and then verify the result by using graphing utility.

Explanation

Given Information:

The provided equations are y=x3lnx,y=0,x=e.

Formula used:

Integration by parts.

When u and v is assumed to be the differentiable functions of x then,

u dv=uvv du

Calculation:

Consider the equation,

y=x3lnx.

Here the lower limit is x=1 and the upper limit is x=e. So, the area of the bounded regions would be,

Area=1ex3lnx dx

Let u=lnx and dv=x3dx, then

dv=x3dx

Apply integral on both sides of the above equation as,

dv=x3dxv=12x2

Thus, v=12x2,

Then differentiate both sides of the equation u=lnx as,

du=d(lnx)du=1xdx

So, du=1xdx

Now apply integration by parts formula.

Substitute u=lnx, dv=x3dx, v=12x2 and du=1xdx in the formula u dv=uvv du as,

x3lnx dx=12x2lnx+12x2(1x) dx=lnx2x2+12x3dx=lnx2x214x2+C

Thus, x3lnx dx=lnx2x2

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