Question

Use definition to evaluate the double integral

$$\begin{gathered} \int {\int }_R{x}^2{y}^{3}dA \\ WhereR=\left\{\right(x,y\left)\right|1\le x\le 3,0\le y\le 2\} \end{gathered}$$

Short answer

The value of the integral is 126

Step-by-step solution

Given information

We are given a double integral

$$\begin{gathered} \int {\int }_R{x}^2{y}^{3}dA \\ WhereR=\left\{\right(x,y\left)\right|1\le x\le 3,0\le y\le 2\} \end{gathered}$$

Use definition to find the value of double integral

We get,

$$\begin{gathered} \sum _{i=0}^2\underset{j=1}{\overset{3}{\sum x^2{y}^3}} \\ =\sum _{i=0}^2(y^3+4y^3+9y^3) \\ =\sum _{i=0}^2\left(14y^3\right) \\ =14\left(1\right)+14\left(8\right) \\ =126 \end{gathered}$$