Reverse the order of integration in the iterated integral ∫(0)^(1)∫(√y)^(2)f(x,y)dxdy

Question

Reverse the order of integration in the iterated integral [latex]\int_{0}^{1} \int_{\sqrt{y}}^{2} f(x, y) d x d y[/latex].

Accepted Answer

The region of integration is sketched in Figure 11. This region is divided into two subregions [latex]\Omega_{1}[/latex] and [latex]\Omega_{2}[/latex]. What happens if we integrate first with respect to [latex]y[/latex] ? In [latex]\Omega_{1}, 0 \leq y \leq x^{2}[/latex]. In [latex]\Omega_{2}, 0 \leq y \leq 1[/latex]. Thus

[latex]\begin{aligned}\int_{0}^{1} \int_{\sqrt{y}}^{2} f(x, y) d x d y &=\iint_{\Omega} f(x, y) d A=\iint_{\Omega_{1}} f(x, y) d A+\iint_{\Omega_{2}} f(x, y) d A \&=\int_{0}^{1} \int_{0}^{x^{2}} f(x, y) d y d x+\int_{1}^{2} \int_{0}^{1} f(x, y) d y d x .\end{aligned}[/latex]

Question 5.2.7: Reverse the order of integration in the iterated integral ∫(...

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