Interchaning limits of integration

Question

Fix $0<a<b$. How would you change the limits of integrations $\int_a^b\int_0^ydxdy$ to something so you were integrating with respect to $y$ first? $\int_0^a\int_x^bdydx$ isn't right. Any help is appreciated

Answer 1

Drawing the integration region helps. $f(x)=\frac{1}{\sqrt{1-x^2}}$ $$\int_a^b \int_0^y f(x) \,\mathrm dx\,\mathrm dy = \int_0^a \int_a^b f(x) \,\mathrm dy\,\mathrm dx +\int_a^b \int_x^b f(x) \,\mathrm dy\,\mathrm dx.$$