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Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770
Textbook Problem

Finding the Volume of a SolidIn Exercises 1–14, use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.

y = 1 2 π e x 2 / 2 ,   y = 0 ,   x = 0 ,   x = 1

To determine

To calculate:

The integral y=12πex22 using Shell method that gives the volume of the solid generated by revolving the plain region about y axis.

Explanation

Given:

y=12πex22,y=0,x=0,x=1

Formula used:

Use Shell method Vertical Axis of Revolution.

V=2πabp(x)h(x)dx

Where,

p(x)= Radius of revolution.h(x)= Height of the shell.dx= Thickness of the shell.

Calculation:

We have y=12πex22,y=0,x=0,x=1

Now, apply the Shell method to calculate the volume of solid.

V=2πabp(x)h(x)dx

Where,

p(x)=xh(x)=12πex22

We now have all the expressions in the terms of x.

Since these expressions = the volume of 1 shell,

We must integrate to sum all the shells between x=0 and x=1,

V=2π01x(12πex22)dxV=2π×12<

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