Q. 18 - Calculus

Question

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$$\begin{gathered} \int {\int }_{\mathfrak{R}}{e}^{xy}dA \\ where \mathfrak{R}=\left\{\left(x,y\right)I1\le x\le 2 and 1\le y\le 3\right\} \end{gathered}$$

Step-by-Step Solution

Answer

The given double integral is difficult to evaluate using iterated integrals.

Step 1. Given Information.

The iterated integral is,

$$\begin{gathered} \int {\int }_{\mathfrak{R}}{e}^{xy}dA \\ where \mathfrak{R}=\left\{\left(x,y\right)I1\le x\le 2 and 1\le y\le 3\right\} \end{gathered}$$

Step 2. Explanation.

Evaluating the first integral with respect to $y$ and treating $x$ as constant.

$$\begin{gathered} \int {\int }_{\mathfrak{R}}{e}^{xy}dA \\ ={\int }_1^2{\int }_1^3{e}^{xy}dydx \\ ={\int }_1^2\left({\int }_1^3{e}^{xy}dy\right)dx \\ ={\int }_1^2{\left[\frac{1}{x}{e}^{xy}\right]}_{y=1}^{y=3}dx \\ ={\int }_1^2\left[\frac{1}{x}{e}^{3x}-\frac{1}{x}{e}^x\right]dx \end{gathered}$$

Now the resultant function $\frac{1}{x}{e}^{3x}-\frac{1}{x}{e}^x$ does not have a simple anti-derivative to evaluate the next step which is integrating with respect to variable $x$. The given double integral is difficult to evaluate using iterated integrals.