Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 2 / iii / Solution

In the original order of integration,In the reversed order, the below-axis contribution is ${\int }_{-1}^{0}y(1+y)dy=-1/6$, and the above-axis contribution is ${\int }_0^{1}y(1-\sqrt{y})dy=1/10$. Their sum is again $-1/15$. A negative answer is possible because the integrand $y$ changes sign; the integral is not the area of the region.