Question

Compute the volume of the intersections of the interiors of spheres:

Answer 1

\[x ^{2}+y ^{2}+z ^{2}=1 (\rho=1)\]

and

\[x ^{2}+y ^{2} +(z-1)^{2}=1 (\rho=?)\]

Answer 2

To find the equation of the second sphere, make the following substitution:
\[\begin{cases}
x=r\cos\theta\sin\phi\\
y=r\sin\theta\sin\phi\\
z=r\cos\phi
\end{cases}\]
which gives
\[\begin{align*}
(r\cos\theta\sin\phi)^2+(r\sin\theta\sin\phi)^2+(r\cos\phi-1)^2&=1\\
r^2\cos^2\theta\sin^2\phi+r^2\sin^2\theta\sin^2\phi+r^2\cos^2\phi-2r\cos\phi+1&=1\\
r^2\sin^2\phi(\cos^2\theta+\sin^2\theta)+r^2\cos^2\phi-2r\cos\phi&=0\\
r^2\sin^2\phi+r^2\cos^2\phi-2r\cos\phi&=0\\
r^2(\sin^2\phi+\cos^2\phi)-2r\cos\phi&=0\\
r^2-2r\cos\phi&=0\\
r-2\cos\phi&=0\\
r&=2\cos\phi
\end{align*}\]

Answer 3

A plot of the intersecting spheres:

Answer 4

To clarify, the green sphere is \(\rho=2\cos\phi\), and the red one is \(\rho=1\).

Define the space by the set \(D\):
\[D:=\left\{(\rho,\theta,\phi)~:~2\cos\phi\le\rho\le1,~0\le\theta\le2\pi,~0\le\phi\le \frac{\pi}{3}\right\}\]
The volume itself is given by
\[V=\int\int\int_DdV=\int_0^{2\pi}\int_0^{\pi/3}\int_{2\cos\phi}^1\rho^2\sin\phi~d\rho~d\phi~d\theta\]

Answer 5

Let me know if you're confused about how to get those limits. I'll leave all the computation to you.