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https://i.stack.imgur.com/uDt4l.jpg

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?

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  • $\begingroup$ 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. $\endgroup$ Nov 28, 2014 at 22:38

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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: $$ \int_0^{\pi/4} \int_0^{\pi/2} \int_{\sec \phi}^{\sqrt 2} \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi $$

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    $\begingroup$ 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 $$\frac1{cos ϕ}?$$ $\endgroup$
    – elogging
    Nov 28, 2014 at 23:12
  • $\begingroup$ Yes, that works too. $\endgroup$
    – Adriano
    Nov 30, 2014 at 7:16

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