I have to determine the volume and the formula for the volume for this spherical cap of height $h$, and the radius of the sphere is $R$:
Two methods: *I just need help setting up the triple integrals
1) Cylindrical
For for this method I am thinking that $\theta$ goes from $0$ to $2 \pi$, $r$ from $0$ to $5$, and $z$ from $R-h$ to $R$ with the integral of $1\text{d}z\text{d}r\text{d}\theta$. However, I'm not sure if this right??
2) Spherical
For this I know that $\theta$ ranges from $0$ to $2 \pi$, but I cannot figure out the range for $\phi$?? I know its from $0$ to some angle where the cap lies ($R-h$), however, I cannot figure it out. Same goes for the range for $\rho$, for this I am assuming it would start at $(R-h)\sec\theta$ to what the outer boundary is? Sorry, I'm completely lost.
I've been working on this problem and trying to set it up for quite some time and have had no luck, and as a last resort, I am asking on here.
To whomever can help me, could you please keep it very detailed?
Thank you.
Best Answer
You can solve this problem using solids of revolution.
Define a circle as $x^2+y^2=R^2$
$\Rightarrow x=\sqrt{R^2-y^2}$
By constructing wide circular cylinders on top of one another, decreasing in size as you move towards the top, one gets
$V=\int\limits_{R-h}^R\pi(R^2-y^2)dy\\=\left.\pi R^2y-\pi\frac{1}{3}y^3\right]_{R-h}^R=\pi Rh^2-\frac{1}{3}\pi h^3$