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Use either a $ CAS $ or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the x-axis.

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

$$

\pi\left(3 \sqrt{19}+\frac{\operatorname{arcsinh}(3 \sqrt{2})}{\sqrt{2}}\right) \approx 45.8621

$$

Applications of Integration

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this question asked us to use caste or table and girls to find the exact area of the surface. What we know we can do is we know we can set this up using our usual formula of two pi times integral from a to B so we know are integral bounds of they said From 0 to 3, we know our low ground zero upper bound us three and then we have square root of ax squared, plus one times square root of one plus acts over spirit of X squared plus one square detox. We know this is equivalent to to pied times integral from 0 to 3. It's not just cleaning this up a bit squared of two X squared plus one DX. Now, if you plug this in using cast, what you know is you end up with pie times three squared of 19 plus inverse sine, which gives you approximately 45 points nine