Iterated Integrals — Question 2

Question 2

Evaluate $$\iint_R e^{2x}\cos y\,dA, \qquad R=[0,\ln 3]\times\left[0,\frac{\pi}{3}\right].$$

Tasks

1. Recognize and use the product structure of the integrand.

2. Evaluate the integral in each order.

3. Verify the exact result by multiplying the two one-variable integrals.

Question 2 – Solution

Strategy. The integrand is a product of an $x$-only factor and a $y$-only factor, so constant rectangular bounds make the two integrations separate.

Step 1: Integrate in $y$ first $$\begin{align*} \int_0^{\ln 3}\int_0^{\pi/3}e^{2x}\cos y\,dy\,dx &=\int_0^{\ln 3}e^{2x}[\sin y]_0^{\pi/3}\,dx\\ &=\frac{\sqrt 3}{2}\left[\frac 12e^{2x}\right]_0^{\ln 3}\\ &=\frac{\sqrt 3}{4}(9-1)=2\sqrt 3. \end{align*}$$

Step 2: Reverse the order $$\begin{align*} \int_0^{\pi/3}\int_0^{\ln 3}e^{2x}\cos y\,dx\,dy &=\int_0^{\pi/3}\cos y\left[\frac 12e^{2x}\right]_0^{\ln 3}dy\\ &=4[\sin y]_0^{\pi/3}=2\sqrt 3. \end{align*}$$ Therefore $$\boxed{\iint_R e^{2x}\cos y\,dA=2\sqrt 3}.$$

Verification Direct factorization gives $$\left(\int_0^{\ln 3}e^{2x}\,dx\right) \left(\int_0^{\pi/3}\cos y\,dy\right) =4\cdot\frac{\sqrt 3}{2}=2\sqrt 3,$$ confirming both orders and the endpoint $e^{2\ln 3}=9$.