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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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 = 1 9 e x x / 3 , y = 0 , x = 0 , x = 3

To determine

To calculate: The area of the region bounded by graphs of the equations y=19xex/3,y=0,x=0,x=3 and then verify the result by using graphing utility.

Explanation

Given Information:

The provided equations are y=19xex/3,y=0,x=0,x=3.

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=19xex/3.

Here the lower limit is x=0 and the upper limit is x=3. So, the area of the bounded regions would be,

Area=0319xex/3dx

Let u=x and dv=ex/3dx, then

dv=ex/3dx

Apply integral on both sides of the above equation as,

dv=ex/3dxv=3ex/3

Thus, v=3ex/3,

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

du=dx

Now apply integration by parts formula.

Substitute u=x, dv=ex/3dx, v=3ex/3 and du=dx in the formula u dv=uvv du as,

19xex/3 dx=19[3xex/3+3ex/3dx]=19[3xex/39ex/3]+C

Thus, 19xex/3 dx=19[3xex/

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