Evaluate \[ \int_0^1\int_y^1\sin(x^2)\,dx\,dy. \]
Solution
The region is \(0\le y\le x\le1\). Reversing the order of integration, \[ \int_0^1\int_y^1\sin(x^2)\,dx\,dy =\int_0^1\int_0^x\sin(x^2)\,dy\,dx =\int_0^1x\sin(x^2)\,dx. \] With \(u=x^2\), \[ \int_0^1x\sin(x^2)\,dx =\frac12\int_0^1\sin u\,du =\frac{1-\cos1}{2} =\boxed{\sin^2\!\left(\frac12\right)}. \]