What is meant by local integration?

Double integration: ∫(limits: upper: 1, lower: 0) ∫(limits: upper: sqrt(y), lower: y) (((x^3)y)/(1-y)) dx dy?

• How to I solve this double integration? ∫(limits: upper: 1, lower: 0) ∫(limits: upper: sqrt(y), lower: y) (((x^3)y)/(1-y)) dx dy I've got as far as (not sure if it's correct) taking out (y/(1-y) as a common factor and integrating once: I = ∫(limits: upper: 1, lower: 0) (y/(1-y) [ (x^4)/4 ](limits: upper: sqrt(y), lower: y) dy I = ∫(limits: upper: 1, lower: 0) (y/(1-y) [ ((y^2)/4) - ((y^4)/4) ] But dunno how to integrate it again... The answer is meant to be 1/16 but I'm not getting that. Can someone solve it and show their steps please? Thank you!

• Answer:

  Are you about this? ∫(y = 0 to 1) ∫(x = y to y^(1/2)) (x^3 y/(1 - y)) dx dy = ∫(y = 0 to 1) (1/4)(x^4 y/(1 - y)) {for x = y to y^(1/2)} dy = ∫(y = 0 to 1) (1/4)((y^2 - y^4) y/(1 - y)) dy = (1/4) ∫(y = 0 to 1) ((y^2 (1 - y)(1 + y) y/(1 - y)) dy = (1/4) ∫(y = 0 to 1) y^3 (1 + y) dy = (1/4) ∫(y = 0 to 1) (y^3 + y^4) dy = (1/4)(1/4 + 1/5) = 9/80. Double check: http://www.wolframalpha.com/input/?i=%E2%88%AB%28y+%3D+0+to+1%29+%E2%88%AB%28x+%3D+y+to+y%5E%281%2F2%29%29+%28x%5E3+y%2F%281+-+y%29%29+dx+dy I hope this helps!