Question

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

$$\begin{gathered} \int {\int }_R(2-3x^2+y^3)dA \\ WhereR=\left\{\right(x,y\left)\right|-3\le x\le 2,3\le y\le 5\} \end{gathered}$$

Short answer

The value of the integral is 113.33units

Step-by-step solution

Given information

We are given an integral

$$\begin{gathered} \int {\int }_R(2-3x^2+y^3)dA \\ WhereR=\left\{\right(x,y\left)\right|-3\le x\le 2,3\le y\le 5\} \end{gathered}$$

Use iterated integral

We get,

$$\begin{gathered} {\int }_3^5{\int }_{-3}^2(2-3x^2+y^2)dxdy \\ ={\int }_3^5{{[2x-x^3+x{y}^2]}^2}_{-3}dy \\ ={\int }_3^5(10-35+5y^2)dy \\ ={{[-25y+\frac{5}{3}{y}^3]}^5}_3 \\ =113.33units \end{gathered}$$