change of variables to find area?

Question

Use a suitable change of variable to find the area of the region R bounded by the graphs of y=x^2, y=4x^2, y=sqr(x), y=(1/2)sqr(x)

Answer 1

Note that all four equations are in form
uy = x²
y² = vx

Solving with respect to x and y we have
x = ³√(u²v)
y = ³√(uv²)

The equtions become
³√(uv²) = (³√(u²v))² u = 1
³√(uv²) = 4 (³√(u²v))² u = 1/4
(³√(uv²))² = ³√(u²v) v = 1
4 (³√(uv²))² = ³√(u²v) v = 1/4

Jacobian of this transformation is 1/3
dx = | 2/3 ³√(v/u) 1/3 ³√(u/v)² | du
dy = | 1/3 ³√(v/u)² 2/3 ³√(u/v) | dv

dS = dx dy = (4/9 - 1/9) dudv = 1/3 du dv

Area = 1/3 ∫ ∫ dudv = 1/3 Δu Δv = 1/3 x 3/4 x 3/4 = 3/16