Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere cut off by the plane and restricted to the first octant.

Question

Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere  cut off by the plane  and restricted to the first octant. What coordinates are you using? 

(Enter cartesian, cylindrical, or spherical.)

With  ,  , ,  , , and  ,Volume =    

Answer 1

Since it's a sphere, lets use spherical coordinates!

∫∫∫dV → ∫dr∫dφ∫dθ r² sin(φ) , with r≥0 , 0≤φ≤pi, 0≤θ≤2pi

this would give the whole sphere. So let's mess with our limits

we want 0≤r≤ Max radius of sphere

0≤φ≤pi/2 (this gives only the upper half of the sphere)

0≤θ≤pi/2 (this gives only the first quadrant)

we're using x=rsinφcosθ , y=rsinφsinθ , z=rcosφ

lets calculate this out for a Max radius = b

∫r²dr ∫sin(φ)dφ∫dθ = b³/6

since a volume of a sphere is 4/3 * pi * r³ and we're looking at the upper half of one of its quadrants, it makes since that our volume equals the total sphere volume divided by 8

= (4/3 * π * r³)/8 = r³/6