21. Multiple Integrals in Curvilinear Coordinates

c. Integrating in Spherical Coordinates

4. Converting & Mixing, Rectangular & Spherical Coordinates

Sometimes it is useful to use spherical coordinates instead of rectangular coordinates when computing an integral. Other times it is best to use rectangular coordinates for one part of an integral and spherical coordinates for another part.

Converting Rectangular to Spherical Coordinates

Sometimes by converting to spherical coordinates, we can simplify the integral greatly.

Computing the mass of a region requires integration of the density function over the region: \[ M=\iiint \delta\,dV=\iiint z(x^2+y^2)\,dV \] The density and volume element in spherical coordinates are: \[\begin{aligned} \delta&=z(x^2+y^2)=(\rho\cos\phi)(\rho^2\sin^2\phi)=\rho^3\cos\phi\sin^2\phi \\ dV&=\rho^2\sin^\phi\,d\rho\,d\phi\,d\theta \end{aligned}\] The cone is: \[ z=\sqrt{x^2+y^2} \quad \Rightarrow \quad \rho\cos\phi=\rho\sin\phi \quad \Rightarrow \quad \tan\phi=1 \quad \Rightarrow \quad \phi=\dfrac{\pi}{4} \] The hemisphere is \[ z=\sqrt{50-x^2-y^2} \quad \Rightarrow \quad x^2+y^2+z^2=50 \quad \Rightarrow \quad \rho=\sqrt{50} \] So the mass is: \[\begin{aligned} M&=\int_0^{2\pi}\int_0^{\pi/4}\int_0^{\sqrt{50}} (\rho^3\cos\phi\sin^2\phi)\rho^2\sin\phi\,d\rho\,d\phi\,d\theta \\ &=\int_0^{2\pi}\int_0^{\pi/4}\int_0^{\sqrt{50}} \rho^5\cos\phi\sin^3\phi\,d\rho\,d\phi\,d\theta \\ &=2\pi \left[\dfrac{\sin^4\phi}{4}\right]_0^{\pi/4} \left[\dfrac{\rho^6}{6}\right]_0^{\sqrt{50}} =2\pi\dfrac{1}{4}\left(\dfrac{1}{\sqrt{2}}\right)^4\dfrac{\sqrt{50}^{\,6}}{6}\\ &=2\pi\dfrac{1}{4}\dfrac{1}{4}\dfrac{50^3}{6} =\dfrac{5^6}{6}\pi \end{aligned}\]

Mixing Rectangular and Spherical Coordinates

Sometimes one integral should be done in rectangular coordinates and another should be done in spherical coordinates, such as in the example below.

We divide the solution into several steps:

1. Find the volume and mass of the hemisphere (with no box cut out).
2. Find the volume and mass of the box that is cut out.
3. Subtract the volume of the box from the volume of the hemisphere to find the volume of the 3D region between them.
4. Subtract the mass of the box from the mass of the hemisphere to find the mass of the 3D region between them.
5. Divide the mass of the region by the volume of the region to get the average density.

Step 1:   First, we use geometry to find the volume of the sphere and use spherical coordinates to find the mass of the sphere. \[ V_\text{hemisphere} =\dfrac{1}{2}\left(\dfrac{4}{3}\pi r^3\right) =\dfrac{2}{3}\pi(8)^3 =\dfrac{1024}{3}\pi \] The density is \(\delta=z=\rho\cos\phi\). So the mass of the sphere is: \[\begin{aligned} M_\text{hemisphere} &=\iiint \delta\,dV =\int_0^{2\pi}\int_0^{\pi/2}\int_0^8 (\rho\cos\phi)\rho^2\sin\phi\,d\rho\,d\phi\,d\theta \\ &=2\pi\int_0^{\pi/2} \sin\phi\cos\phi\,d\phi\int_0^8\rho^3\,d\rho \\ &=2\pi\left[\dfrac{\sin^2\phi}{2}\right]_0^{\pi/2} \left[\dfrac{\rho^4}{4}\right]_0^8 =2\pi\left(\dfrac{1}{2}\right)\dfrac{8^4}{4} =1024\pi \end{aligned}\]

Step 2:   Using rectangular coordinates, we can easily find the volume of the box: \[ V_\text{box}=6\cdot6\cdot3=108 \] And the mass of the box is \[\begin{aligned} M_\text{box} &=\iiint \delta\,dV=\int_{-3}^3\int_{-3}^3\int_0^3 z\,dz\,dy\,dx \\ &=36\left[\dfrac{z^2}{2}\right]_0^3 =162 \end{aligned}\]

Step 3:   Next we subtract the volume of the box from the volume of the sphere: \[ V_\text{region}=V_\text{hemisphere}-V_\text{box} =\dfrac{1024}{3}\pi-108\approx964.3 \]

Step 4:   Next we subtract the mass of the box from the mass of the hemisphere: \[ M_\text{region}=M_\text{hemisphere}-M_\text{box} =1024\pi-162\approx3055.0 \]

Step 5:   Finally, we divide the total mass of the region by the total volume of the region: \[ \delta_\text{ave}=\dfrac{M_\text{region}}{V_\text{region}} =\dfrac{1024\pi-162}{\dfrac{1024}{3}\pi-108}\approx3.168 \]

The density \(\delta=z\) ranges from \(0\) to \(8\). So \(\delta_\text{ave}\approx3.168\) is within this range.