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Let $ S $ be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume $ V $ of $ S $. Sketch a typical approximating shell. What are its circumference and height? Use shells to find $ V $.

$\begin{aligned} V &=\frac{\pi}{15} \end{aligned}$

Applications of Integration

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Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

okay. They've told us to sketch the show. Well, first thing is sketched the function. We know it would look a little something like this, which means the shell would look something like this, perhaps. Okay, Now we know the circumference is to part times the radius, therefore see, is two part times X. Remember, height is the wide distance, which in this case, is gonna be acts times X minus one squared. So the formula is from our bounds. In this case, 01 to pod axe times acts, which is X squared, then times X minus one squared. D backs. Now remember, we can pull up the constant so two pies, the constant it can be pulled out. And we can actually expand this using the distribution method. Now that we have this, we know we can integrate using the power rule, which means we increased the exponents by one. And then we divide by the new exponents. So extra the fourth becomes X to the fifth, divided by five. Plug in one. We have to pie once the fifties. Just 1/5 once. The fourth is just 1/4. One cube, just just one third remembers dividing by all of these on the bottom, which gives us pi over 15. So remember, our volume is power over 15 sees two pi axe and capital H is X times X minus one square.