Find the volume of sphere x2+y2+z2=a2
Answers
Answer:
Taking the equation for the cylinder I completed the square to find (x−a2)2+y2=a24 and the sphere clearly has radius a and is centered at the origin. Now to solve this question we express the radius in terms of theta using the equation for the cylinder (giving r=acosθ) and then we solve for z in the sphere's equation giving z=a2−x2−y2−−−−−−−−−−√=a2−r2−−−−−−√ and setup the integral as follows (multiplying by 4 since we only consider the first octant but the total area is in 4 octants):
4∫π20∫acosθ0a2−r2−−−−−−√rdrdθ
This is the part I don't understand, why exactly do we express the radius in terms of the cylinder and then why do we solve for z in terms of the sphere and integrate that? Clearly it gives the volume contained but I can't fathom how.