Volume between a sphere and coneI'm having some problems finding the volume between a sphere of radius 5 and the cone z = − x 2 + y 2 .The bounds I got for spherical coordinates are 0 to 2 π for θ, 3 π / 4 to π for ϕ, 0 to 5 for r, however I'm not sure if these are correct. My thought process is that if you let say x or y = 0 in the equation for a cone, you get an inverted modulus graph which is where I got the bounds for ϕ.For cylindrical coordinates I got bounds of 0 to 2 π for θ, 0 to 5 for r and 0 to 25 − r 2 for z and after evaluating the triple integral I got a volume of 125/3.Could someone run me through their thought process and show me where I went wrong?

Question

Volume between a sphere and coneI&#039;m having some problems finding the volume between a sphere of radius 5 and the cone   z  =  −            x      2        +          y      2      .The bounds I got for spherical coordinates are 0 to   2  π for   θ,   3  π      /    4 to   π for   ϕ, 0  to 5 for r, however I&#039;m not sure if these are correct. My thought process is that if you let say x or   y  =  0 in the equation for a cone, you get an inverted modulus graph which is where I got the bounds for   ϕ.For cylindrical coordinates I got bounds of 0 to   2  π for   θ, 0 to 5 for r and 0 to       25    −          r      2       for z and after evaluating the triple integral I got a volume of 125/3.Could someone run me through their thought process and show me where I went wrong?

Adalyn Pitts · Accepted Answer

Step 1For spherical coordinates, your bounds look fine.For cylindrical coordinates, though, you have   0  ≤  r  ≤  5; that is wrong. r ranges from 0 to the value that r takes along the circle of intersection of the cone and the sphere.Step 2I got       125    3    (  2  −      2    )  π for the value of the integral.

Find the volume between a sphere of radius 5 and the cone z=-sqrt{x^2+y^2}.

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