How do I evaluate this double-integral?

How do I evaluate this double-integral?

The problem statement is as follows: Evaluate iint xydydx over the region bounded by the curves x y2, x 2 - y, y 0 and y 1. My confusion is with regards to how the question is to be interpreted. I have tried the following two methods: Plot the curves on the graph and find the coordinates of the points enclosing the region of interest - doing so, I obtained the limits: x goes from 0 to 2 derived from the plot, y goes from 0 to 1 as stated in the question. Thus, my integral would look like so: int_01 int_02xy dydx and evaluate to 1. Treat the integral like so:int_01 int_y22-yxy dydx in which case, the integral evaluates to: frac 144 y2 - 4y - y4 Which of these is the right way to go and how do I identify what is expected of me in solving the question

Do a drawing of the horizontal parabola and the straight lines. We have: int_01int_y21xy,mathrm dx,mathrm dyint_12int_02-xxy,mathrm dy,mathrm dxfrac12int_01y1-y4,dyfrac12int_12xleft2-x2-02right,dx frac14-frac112frac12int_12left4x-4x2x3right,dxfrac16frac12left24-1-frac438-1frac1416-1right frac163-frac286frac158frac472-1124524frac924frac38 Observe that your integration domain is irregular and thus we had to split it into two regular regions...Check the arithmetic above.

It is a good idea to draw the region https:www.desmos.comcalculator7v8tu9svjk Your second attempt is good begineqnarray int_01 y left int_y22-y x dx right dy . endeqnarray Now the inner integral gives begineqnarray int_y22-y x dx left fracx22 right_y22-y. endeqnarray I will let you complete the problem from here. Edit : This will give begineqnarray int_01 fracy2 left 4-4yy2-y4 rightdy endeqnarray Don Antonios value colorredfrac38 agrees with Wolfram alpha http:www.wolframalpha.cominputiint_05E1int_7By5E27D5E7B2-y7Dxydxdy and the value I have calculated in my notebook.