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Numerade Educator

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Problem 15 Medium Difficulty

The given curve is rotated about the y-axis. Find the area of the resulting surface.

$ y = \frac{1}{3} x^{\frac{3}{2}} $ , $ 0 \le x \le 12 $

Answer

$$
\frac{3712}{15} \pi
$$

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Video Transcript

this question states that the given curve is rotate about the y axis and we know we need to find the area. What we know we need to do is consider the fact that we're looking at bounds between zero and 12. Exit between zero and 12 has given the problem. We also know we have to pile on the outside. Their formula essentially is to pi times the integral from a to b times X squared of one plus f of x f prime of acts which essentially the derivative times d of acts so plugging into this formula. And again, this is listed in the textbook as well. With examples, we know we can simplify. Now what we know we can do is there's a lot going on here with square root. So I would recommend using u substitution. If you is four plus tax, we know acts is you minus four. This helps us because now we know we can write this out. Integrate this. We integrate. Remember, we know what us and we know what X is. We know we integrate by increasing exported by one dividing by the new expert is you can see I'm dividing by the new exploding a five over too. Now you can plug in. Remember, you're always doing your upper bounds minus your lower bounds in this context, and I end up with 3712 pi divided by 15.