How do you find the volume of a cylinder using only spherical polar coordinates?
Let R be radius of cylinder. Consider the part of the cylinder where a line through the origin passes through the walls of the cylinder. That’s all of the cylinder except two cones at the top and bottom. maximum r for any elevation theta = R / cos theta theta range is +/-arctan (h/2R) Volume integral dV = r^2 cos theta dtheta dr dphi Integrate succssively within the ranges dphi, dr, dtheta dphi gives 2pi, dr gives r^3/3 which becomes (1/3)(R^3/cos^3 theta) Now you have integral 2/3 pi R^3/(cos^2 theta) dtheta.
