R is the region bounded by y=x^2 and y=4. evaluate the double integral of f(x,y)=6x^2+2y over R
After drawing the region I was wondering if I could just work with the first quadrant and then double my solution, because both y=x^2 and y=4 are even functions so my question is does my solution work? If so would my very first line be correct? Oh and I'm not sure how to write what I'm integrating between so when I put ∫[a,b]f'(x)dx thats f(a)-f(b).
The Attempt at a Solution
∫[4,0]∫[y^1/2,-y^1/2]6x^2+2y dxdy = 2*∫[4,0]∫[y^1/2,0] 6x^2+2y dxdy
following it through
= 2*∫[4,0] [[2(y^1/2)^3+2*y*y^1/2]-] dy
= 2*∫[4,0] 4*y^3/2 dy