I want to find the volume of a sphere cap using the solids of revolution method. Let the sphere have radius $r$ and the cap have height $h$. Then the volume of the cap is given by $\pi h^2\left(r-\frac{1}{3}h\right)$.
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I tried to derive the expression as follows: the equation for a circle with centre $(0,0)$ is $x^2+y^2=r^2\implies x=\sqrt{r^2-y^2}$ So the volume = $$\int\limits_{r-h}^r \pi(r^2-y^2)\operatorname{d}y=\left[\pi r^2y-\frac{1}{3}\pi r^2y^3\right]_{r-h}^r=\pi r^2h-\frac{1}{3}\pi r^2h^3$$ However, this solution is wrong. Where is the error in my working?
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Best Answer
The source of error is in your integration.
The integration should yield the following result $$\int\limits_{r-h}^r \pi(r^2-y^2)\operatorname{d}y=\left[\pi r^2y-\frac{1}{3}\pi y^3\right]_{r-h}^r=\pi rh^2-\frac{1}{3}\pi h^3$$
Once you've corrected this part you'll get the right answer.