A very simple definite integral problem Find the volume of the body of revolution of the semi ellipse X & sup2 / 9 + Y & sup2 / 4 = 1 (Y > = 0) rotating around the x-axis

A very simple definite integral problem Find the volume of the body of revolution of the semi ellipse X & sup2 / 9 + Y & sup2 / 4 = 1 (Y > = 0) rotating around the x-axis

V=∫[-3,3]0.5*π*y*ydx=∫[-3,3]π*2*（1-x*x/9）dx=8π
Reason: because y ≥ 0, the cross-sectional area s = 0.5 * π * y * y
So the volume of the rotating body is SDX = 0.5 * π * y * YDX

How to determine the upper and lower limits of a definite integral after substitution? For example, if 0 = cost, is t π / 2 or - π / 2 or other? How to determine

Arccosx has a domain of - 1 to 1 and a range of 0 to π