I am asked to find the double integral of 2xy dy dx over a region:
I have found the equation of some lines.
Between (0,0) and (4,4), y=x
Between (5,0) and (4,4), y = -4x + 20
I am confused as to why I have to split the integral.
Why can't the limits for the integral with respect to x go from 0 to 5, and the integral with respect to y go from y=x to y=-4x + 20?
Thank you for your time, I appreciate any help, :)
When the outer integral is the one with respect to $x$, the inner integral with respect to $y$ should be $\int_{y=\alpha(x)}^{\beta(x)}$, where $y=\alpha(x)$ is the equation of the curve which forms the lower boundary of the region, and $y=\beta(x)$ is the equation of the curve which forms the upper boundary.
In this case, the line $y=0$ is the lower boundary, so $\alpha(x)=0$, while the lines $y=x$ and $y=20-4x$ form different parts of the upper boundary, so $$ \beta(x) = \begin{cases} x,& 0 \le x \le 4,\\ 20-4x,& 4 < x \le 5. \end{cases} $$ Because of the division into cases in the function $\beta$, you have to split the computation into two integrals (one of which has $x$ as the upper bound in the inner integral, while the other one has $20-4x$ as the upper bound).
If you want to do the integral in one stroke, you can instead take the $y$ integral on the outside (from 0 to 4), and let the bounds on the inner $x$ integral go from the left boundary curve $x=y$ to the right boundary curve $x=(20-y)/4$.