Neatly sketch the region of integration and rewrite the double integral using dxdy.

Question

Definite Integral Bounds Upper=4 Lower=4, second Definite Integral Bounds Upper=sqrt(x) Lower= y=x/2

f(x,y)dydx

Any leads my fellow mathematicians?

Answer 1

William is correct, we need lower = 0 (√x = x/2 -> x = x^2/4 or x^2/4 - x = 0 or 1/4x(x-4) = 0; or x = 0, 4

lower: y >= x/2 -> x <= 2y (the new upper bound)

upper: y <= √x -> y² <= x, or x >= y² (the new lower bound)

When x = 0, y = x/2 = 0

When x = 4, y = x/2 = 4/2 = 2

∫₀² ∫_y^2 ^2y f(x,y)dx dy