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Problem 38 Hard Difficulty

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ y = -x^2 + 6x - 8 $ , $ y = 0 $ ; about the x-axis

Answer

$\frac{16}{15} \pi$

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Video Transcript

I'm gonna be using this method for this problem. Which means I have V is pi. Times are bounds from 2 to 4 times are why? Which is negative ax squared plus six X minus eight. Now, remember, one crucial piece of information. This has to be squared. What's in parentheses? Haas To be squared times, DX. Which means now, before we take the integral, I would recommend doing the extra step of distributing and foiling to make this easier to recognize the separate terms. Otherwise, it gets a lot more difficult when you're trying to integrate. Now we can integrate. We're gonna be using the power method, which means we increased the expert it by one and then we divide by the new exponents. As you can see over here 1/5 x to the fifth. That's a good example of this. If you have a coefficient, please take that into account as well. When you're multiplying by whatever you're dividing by, if you're dividing by the new exponents plugging end, we end up with plug in four minus plugged in to via 16. Divide by 15. It's a 16 pi over 15