Q. 36 - Calculus

Question

Evaluate each of the integrals in exercise 33-36 as iterated integrals and then compare your answers with those you found in exercise 29-32

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

Step-by-Step Solution

Answer

The value of the integral is $\frac{104}{3}$

Step 1: Given information

We are given an integral 

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

Step 2: Use iterated integral

We get,

$$\begin{gathered} {\int }_0^2{\int }_1^3{x}^2{y}^{3}dxdy \\ ={\int }_0^2{{\left[\frac{x^3{y}^3}{3}\right]}^3}_{1}dy \\ ={\int }_0^2\left[\frac{26y^3}{3}\right]dy \\ ={{\left[\frac{26}{12}{y}^4\right]}^2}_0 \\ =\frac{104}{3} \end{gathered}$$