Iterated Integrals — Question 3

Question 3 – Solution

Strategy. Treat $x$ as constant in the inner integral; the factor $x$ is exactly the derivative needed for $e^{xy}$ with respect to $y$.

Step 1: Efficient order For $x\ne 0$, $$\int_0^2xe^{xy}\,dy=[e^{xy}]_0^2=e^{2x}-1.$$ At $x=0$ the original integrand is identically zero and the same expression gives $e^0-1=0$, so the formula extends continuously. Hence $$\begin{align*} I&=\int_0^1(e^{2x}-1)\,dx =\left[\frac 12e^{2x}-x\right]_0^1\\ &=\boxed{\frac{e^2-3}{2}}. \end{align*}$$

Step 2: Reversed order The rectangle permits $$I=\int_0^2\int_0^1xe^{xy}\,dx\,dy.$$ For $y\ne 0$, integration by parts gives $$\int xe^{xy}\,dx=\frac{e^{xy}(xy-1)}{y^2}+C,$$ so the inner result is $$\frac{e^y(y-1)+1}{y^2}.$$

Step 3: Efficiency and verification The reversed expression has a removable singular appearance at $y=0$ and requires another nonobvious antiderivative. The original order collapses immediately by substitution. Continuity of $xe^{xy}$ on the rectangle guarantees by Fubini’s theorem that either completed order has the same value.