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Problem 37 Hard Difficulty

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

$ y = \sin^2 x $ , $ y = 0 $ , $ 0 \le x \le \pi $ ; about $ y = -1 $

Answer

$\frac{11}{8} \pi^{2}$

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Video Transcript

the problems is use a computer algebra system to find the exact volume of the solid obtained by rotating the region Bounded by the given curves about the specified line by equals two. Science square eggs. Why equals to zero? Zero? Should be less than or equals two. X. is a dinner equals true by that is it should be Between 0 to fight About by equals 2 -1. So how do we do? So, first of all we will just use a computer and the breath system. So this is the graph we have obtained from computer the address system. So this is the basically grass. So we asked going to rotate this. Yeah here like this in this way. Right. So what happens is according to the graph we can see that we can see from that and some grass. Okay. Yes we can see. Arabs is nothing but after rotation. Okay, this will be science square eggs minus of minus money. Right? And now Small Rx will be one. There's gonna be zero minus minus one. Six people. So this will be a quest towards nothing. But it will be one. No from the wash your method. Mhm volume of these holidays. Mhm Yeah. Volume. Oh solid is given by we equals two. Integration of 8 to be limited. Bye. Capital. Our eggs squared minus small rxc square See it should be function So this should have square tick. This will be square be it. All right. So we have already found out that after rotation what is happening here. So this will be given by the board. This movie by is taken common. And the limits limits us zero to buy as it is given in the problem that is it should be between 0-5. So now let's will be one and it will be signed square. It's less one who is with minus one square. Okay, this ruby. It is, yeah. Okay. So now by is again taking common. So the other two by again, it will be told signed to the powerful eight plus. Okay, so it should be one again. Mhm Plus, it should be two sides square. It's Mhm. Mhm. And it will be one square means one minus one. This will give us good. This will give us bye. My life. 0 to buy. This will be assigned to the power for it plus to sign square eggs the incident. Mhm. Hey, come on. Okay. So using using no using computed no algebra system. Yeah, we have, Yeah. Mm hmm. B equals two. 11 by 85 square. So we are we are doing a thing but we are just integrating it here and we are getting the values of this is the value of the volume of this go. I hope you're mr Defonseca. Thank you for watching. Yeah.