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Find the exact area of the surface obtained by rotating the curve about the x-axis.

$ y^2 = x + 1 $ , $ 0 \le x \le 3 $

$$

\frac{1}{6} \pi(27 \sqrt{27}-5 \sqrt{5})

$$

Applications of Integration

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this question asked us to find exact area of the surface by rotating the curb about the X axis. As it stated, What we know we need to do is we know we need to first off, figure out the bounds. They've actually given this to us of exes between zero and three. This means zero and three are about. Now we know we pull out to pie because we know we're gonna be rotating it as it's stated in the problem. And then we know that we have Why squared is X plus one. What we know this means is that if axes y squared minus one, then D axe over D. Y is equivalent to why this is critical here. Because now what we know we have is remember when we're writing this were essentially plugging in using X. So that is the same thing. That's why you can consider this to be. And then we know that we need to integrate. So in order to do that, we know we're going to be We know we're going to be using the for the multiplication pattern. Essentially distribution eight times people see, is a B plus a C So it's from 0 to 3 to pie X plus five over four de axe. We know we're going to be integrating this. We use the power rule of increasing the exploded by one dividing by the new exponents. That's how you integrate. And then we plug in and what we end up with is pi. Over six time 17 squared of 17 minus five squared of five.