**VOLUME **

**SHELL **

**METHOD **

DONE BY: Leen Azzam

**VOLUME **

**SHELL **

**METHOD **

DONE BY: Leen Azzam

Introduction

The shell method is a technique for finding the volumes of solids of revolutions. It considers vertical slices of the region being integrated rather than horizontal ones, so it can greatly simplify certain problems where the vertical slices are more easily described.

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FORMULA

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** 1**

Consider the Diagram below:

A region RRR is bounded above by the graph of y=\cos xy=cosxy, equals, cosine, x, bounded below by the graph of y=\sin xy=sinxy, equals, sine, x, and bounded on the left by the yyy-axis. Rotating region RRR about the yyy-axis generates a solid of revolution SSS. Find an expression for the volume of SSS.

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draw one representative shell for the solid generated by rotating region RRR about the yyy-axis.

**STEP 1 **

To find the limits of integration, we must find the points of intersection of y=cos x and y=sin x. The two graphs intersect x=Ï€/4. We must now find r and h. In this case, r=x, and h=cosxâˆ’sinx, because of cos x â‰¥ sin x on the interval (0, Ï€/4).

**STEP 3**

The volume can be found using shells. The volume of a solid of revolution on the interval [a,b]when using shells, can be expressed

where r is the radius of rotation and h is the height of a typical shell.

**STEP 2 **

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The volume of the solid of revolution is

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**2**

Consider the Diagram below:

Find the volume of the solid obtained by rotating about the y-axis the region bounded by Y =2x^2 -x^3 and y=0.

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From the sketch, in the Figure, we see that a typical shell has a radius x, circumference 2Ï€x, and height f(x)= 2x^2 - x^3. So, by the shell method, the volume is:

**STEP 1 **

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**THANK YOU**