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Let $ S $ be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of $ S $. Do you think this method is preferable to slicing? Explain.

$\begin{aligned} V &=2 \pi \end{aligned}$

02:52

Wen Z.

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 3

Volumes by Cylindrical Shells

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Lectures

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Let $ S $ be the solid obt…

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Let $ V $ be the volume of…

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Let $V$ be the volume of t…

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Use the method of cylindri…

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Let $W$ be the volume of t…

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Use the shell method to fi…

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given a figure that looks something like Thus we know that if we sketch the cylindrical shell, it probably looks something like this. Why is sign of X squared as given in the problem? Therefore, what we know, given this is that if the circumference is to pie attacks, the height is sign of ax squared. Those are two of the things as part of the answer thin. The volume is from our bounds zero to square root of pi to pi X times sign of ax squared As I just said earlier, we're just multiplying these two, which I announced. Now we know that when we integrate we we can first pull out our Constance so pi times the integral from zero to pie of sine of t DT Remember, the integral of sine is pretty straightforward. It's simply negative co sign of tea and then we all said the pile on the outside from zero to pie, which gives us now we can plug in pied times negative coastline of pi plus Ron which is simply pipas pie which is to pie

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