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Problem 35 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 = \sin^4 x $ , $ 0 \le x \le \pi $ ; about $ x = \frac{\pi}{2} $

Answer

$\frac{1}{12} \pi^{3}$

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Video Transcript

Okay, The first thing we know that the circumference is two pi times pi over to minus X. We know the height is gonna be sine squared of acts minus sign to the fourth of acts. Therefore, the volume is gonna be integral from zero to pi over too of pi over to minus acts. Because remember, it's R H as we've specified over here, and we have circumference. And then we have time sign sport of X minus sign to the fourth of acts times, DX. Now, as a CZ, this problem has specified we know that we are going to be integrating this, which means we're gonna end up with four separate parts that could be broken up. But because it's saying uses a computer algae braided system. We can actually just use a calculator. And remember, we want our answer exact, which means that instead of having 3.4 you bought one for estimate is our value for pi. We just want to leave it in terms of pie, which means we just want to write pi cubed over 32 which means you can plug this into your calculator. But remember whatever the value is with pie. You can't estimate it because 3.14 is close to the exact five pile, but it's not exact the questions that exactly need to write Pied Cube Divide by 32.