Q. 40 - Calculus

Question

Evaluate each of the double integral in the exercise 37-54 as iterated integrals   

$$\begin{gathered} \int {\int }_R(x-e^y)dA \\ Where R=\left\{\right(x,y\left)\right|-3\le x\le 2,-2\le y\le 2\} \end{gathered}$$

Step-by-Step Solution

Answer

The value of integral is $e^2-e^{-2}$

Step 1: Given information

We are given an integral 

$$\begin{gathered} \int {\int }_R(x-e^y)dA \\ Where R=\left\{\right(x,y\left)\right|-3\le x\le 2,-2\le y\le 2\} \end{gathered}$$

Step 2: use iterated integral

We get,

$$\begin{gathered} {\int }_{-2}^2{\int }_{-3}^2(x-e^y)dxdy \\ ={\int }_{-2}^2{{[\frac{x^2}{2}-x{e}^y]}^2}_{-3}dy \\ ={\int }_{-2}^2(-\frac{5}{2}+e^y)dy \\ ={{[-\frac{5}{2}y+e^y]}^2}_{-2} \\ =e^2-e^{-2} \end{gathered}$$