Question

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32

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

Short answer

The value of integral using iteration method is$$\begin{gathered} \int {\int }_R{x}^{2}ydA \\ WhereR=\left\{\right(x,y\left)\right|-1\le x\le 0,0\le y\le 2\} \end{gathered}$$is$\frac{2}{3}units$

Step-by-step solution

Given information

We are given an integral$$\begin{gathered} \int {\int }_R{x}^{2}ydA \\ WhereR=\left\{\right(x,y\left)\right|-1\le x\le 0,0\le y\le 2\} \end{gathered}$$

Evaluate the integral

We get,

$$\begin{gathered} {\int }_0^2{\int }_{-1}^0{x}^{2}ydxdy \\ {\int }_0^2{{\left[\frac{x^{3}y}{3}\right]}^0}_{-1}dy \\ ={\int }_0^2\frac{y}{3}dy \\ =\frac{1}{6}{{\left[y^2\right]}^2}_0=\frac{2}{3} \end{gathered}$$