Q. 35 - Calculus

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 

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

Step-by-Step Solution

Answer

The value of integral is 0

Step 1: Given information

We are given an integral $$\begin{gathered} \int {\int }_{R}x{y}^{3}dA \\ Where R=\left\{\right(x,y\left)\right|-2\le x\le 2,-1\le y\le 1\} \end{gathered}$$

Step 2: Use iteration method to evaluate integral

We get,

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