How to Calculate Volume Using Triple Integrals for a Sphere and Planes?

PhysDrew
Messages
64
Reaction score
0

Homework Statement


Use a triple integral to calculate the volume of the solid enclosed by the sphere
x^2 + y^2 + z^2=4a^2 and the planes z=0 and z=a


Homework Equations


Transform to spherical coordinates (including the Jacobian)


The Attempt at a Solution


I'm stuck, as the radius of the sphere is not constant through the area of integration due to the plane z=a. It looks like I should split this integral up, but I'm just not sure on how to do it. It looks like the angle rho is (pi)/3 when the radius of the sphere(2a, from the origin) hits the z=a plane. Please help!
 
Physics news on Phys.org
Yes, for \phi< \pi/3 the upper boundary is the z= a plane. So, going up from the origin to z= a, then over to the angle \phi, you have a right triangle with angle \phi and near side of length a. \rho is the length of the hypotenuse of that right triangle. \cos(\phi)= a/\rho so
\rho= \frac{a}{\cos(\phi)}= a \sec(\phi).
 
HallsofIvy said:
Yes, for \phi< \pi/3 the upper boundary is the z= a plane. So, going up from the origin to z= a, then over to the angle \phi, you have a right triangle with angle \phi and near side of length a. \rho is the length of the hypotenuse of that right triangle. \cos(\phi)= a/\rho so
\rho= \frac{a}{\cos(\phi)}= a \sec(\phi).
Oh good so I was right, thanks. So should I integrate with rho being between 0 and 2a, with phi being between pi/2 and pi/3, and then...well I get stuck there. I've got that little cone bit in the middle to go and I don't know how to get him (or her). Sorry I'm just not getting this one
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top