/sci/ - Science & Math

>>10023574
The integral in question is in the y direction first, then in the x direction. How i like to think about it, is when integrating in the y direction, you go from bottom to top, that is in the case of this curve, you integrate starting from the curve y=x^2 all the way up to the curve y=2x. Then, in the x direction, it is from left to right, so in this case, from x=0 to x=2, while constrained to the area from the previous integration.

Hence, to switch up the order of integration, what you want to do is first: invert the formulas for y so that you get x= sqrt y and x=y/2. Now you want to integrate from left to right, but this time, you start from the curve x=y/2 and finish at x= sqrt y, so the integral has those boundaries. Now, you want to integrate from the bottom up while constrained to these boundaries. The boundaries are discovered by simply solving the equation 2x=x^2 which is x=0 and x=2, corresponding to y=0 and y=4. Hence, the integration in the y direction is from 0 to 4