Craig Garrett
08/31/2023 · High School
Consider the following curve. \[ x=\frac{1}{3}\left(y^{2}+2\right)^{3 / 2}, 1 \leq y \leq 3 \] Set up an integral in terms of \( y \) that can be used to find the area of the surface \( S \) obtained by rotating the curve about the \( x \)-as Find the exact area of the surface obtained by rotating the curve about the \( x \)-axis. Need Help?
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The surface area \( S \) can be found by evaluating the integral:
\[
S = \frac{2\pi}{3} \int_1^3 \frac{1}{3}(y^2 + 2)^{3/2} (y^2 + 1) \, dy
\]
The exact value of \( S \) depends on the result of this integral.
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