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Problem 23 Medium Difficulty

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = \cos^4 x $ , $ y = -\cos^4 x $ , $ \frac{-\pi}{2} \le x \le \frac{\pi}{2} $ ; about $ x = \pi $

Answer

a) 4$\pi \int_{-\pi / 2}^{\pi / 2}(\pi-x) \cos ^{4} x d x$
b) $\frac{3 \pi^{3}}{2} \approx 46.50942$

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Video Transcript

okay, If we're using the cylindrical method and we're rotating about a vertical axis, we know we're gonna have a d x integral, not a d Y Integral, which was Everything is gonna be in terms of X, which means the height of the integral is gonna be to co signed to the fourth of acts. We know our is gonna be prime eyes acts. Remember, it's going from negative power to two pi over too. Therefore, what we know is using the general formula for Volume two pi times interval from A to B again, our bounds our night of power to two pi over too. If Air Putnam one auras times h times, DEA backs This can be simple for you to be four pi Pull out our constance on the outside just to make it easier for yourself because now it's a lot easier to plug into the calculator, which is what we're gonna be doing in part B. Okay, now that we have done this, we know Part B is to plug into the calculator, which means if you do this, you end up with 46 point 0.50942 and again. You're going to need some sort of graphing calculator. Sent me more advanced calculator than scientific and order to have the functions order to plug this in.