Polar Coordinates and Change Order of Integration; Volume of an Ellipsoid; Change of Variables on Continuous Function of One Variable

Assuming r, θ are the polar coordinates, change the order of integration:
∫-pi/2-->pi/2 ∫0-->a cos θ f(r, θ ) dr dθ

Find the volume of the ellipsoid: x^2/a^2 + y^2/b^2 + z^2/c^2 ≤ 1

Let a and b be any numbers such that a^2 + b^2 =1 and f(x,y) be a continuous function of one variable. Perform the change of variables :
{u =ax +by
{v=bx - ay

Polar Coordinates and Change Order of Integration; Volume of an Ellipsoid; Change of Variables on Continuous Function of One Variable are investigated. The solution is detailed and well presented.