# Triple integration using spherical coordinates

Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere $$x^2+y^2+z^2 ≤ 2$$ cut off by the plane z=1 and restricted to the first octant. Note: In your answer(s), type theta, rho, and phi in place of θ, ρ and ϕ, as needed

Progress: I've managed to find some of the boundaries for integration using spherical coordinates, but for some reason following examples in textbooks I cannot arrive at the rest. Can someone kindly explain?

• Your $\phi$ and $\rho$ limits are not right. You need to think about the region above the plane $z=1$ and inside the sphere. Draw a picture and think about how you represent this in spherical coordinates. Nov 28, 2014 at 22:38

Converting to spherical coordinates, our boundaries are: $$x^2 + y^2 + z^2 = 2 \iff \rho^2 = 2 \iff \rho = \sqrt 2 \\ z = 1 \iff \rho\cos\phi = 1 \iff \rho = \sec \phi$$ Equating the two boundaries in order to find where they intersect, observe that: $$\sec \phi = \rho = \sqrt 2 \iff \cos \phi = \frac{1}{\sqrt 2} \iff \phi = \frac{\pi}{4}$$ So the desired triple integral is: $$\int_0^{\pi/4} \int_0^{\pi/2} \int_{\sec \phi}^{\sqrt 2} \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi$$
• great, thank you! you are awesome! one question though: is there a way to do this question without using sec? in my current course, we have yet to come across examples and practice problems which require sec, so i'm wondering if there's a way to avoid using it. or would it be too cumbersome to use $$\frac1{cos ϕ}?$$ Nov 28, 2014 at 23:12