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Problem 26 Hard Difficulty

Use a $ CAS $ to find the exact area of the surface obtained by rotating the curve about the y-axis. If your $ CAS $ has trouble evaluating the integral, express the surface area as an integral in the other variable.

$ y = \ln (x + 1) $ , $ 0 \le x \le 1 $

Answer

$$
\pi\left(\sqrt{2}-\sinh ^{-1}(1)+\sinh ^{-1}(2)-2 \tanh ^{-1}(\sqrt{2})+2 \tanh ^{-1}(\sqrt{5})\right) \approx 3.69499
$$

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Video Transcript

this question asked us to find the exact area of the surface obtained by rotating the curve about the Y axis. What we know is that we're going to be looking at the fact that we have if X plus One is eating the Y than X is you the wife minus one, which means X prime or the derivative is each the Why? Because this is just a constant. So this interest derivative would just be zero. So now we have from our bounds of a to be lowered upper bound zero to natural log of two again, we get this because we pulled out the natural log by taking ecause we know eat times natural log simply cancels out and gives us one plugging this into your software. We end up with 3.69