Q. 38 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The value of integral is $\frac{80}{3}units$

Step 1: Given information

We are given an integral 

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

Step 2: Use iterated integral

We get,

$$\begin{gathered} {\int }_{-2}^2{\int }_{-3}^2{y}^{2}dxdy \\ ={\int }_{-2}^2{{\left[x{y}^2\right]}^2}_{-3}dy \\ ={\int }_{-2}^{2}5y^{2}dy \\ ={{\left[\frac{5}{3}{y}^3\right]}^2}_{-2}dy \\ =\frac{80}{3} \end{gathered}$$