I need to calculate the area of the surface side obtained by rotating around the $z$-axis the graph of the function $y=\frac{z^2}{2}$, where $z\in [0,1]$. How can I do?
How to calculate area
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integration
area
1 Answers
1
Compute the integral $$S=2\pi\int_0^1y\sqrt{1+(y')^2}\text{d}z=2\pi\int_0^1 \frac{1}{2}z^2\cdot\sqrt{\strut 1+z^2}\text{d}z$$
You can either substitute $z=\sinh u$ or use Euler substitution $\sqrt{\strut 1+z^2}=t-z$.