Q.33 - 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

$$\begin{gathered} \int {\int }_{R}xydA \\ Where R=\left\{\right(x,y\left)\right|0\le x\le 2,1\le y\le 4\} \end{gathered}$$

Step-by-Step Solution

Answer

The value of integral is $$\begin{gathered} \int {\int }_{R}xydA \\ Where R=\left\{\right(x,y\left)\right|0\le x\le 2,1\le y\le 4\} \end{gathered}$$ is 15units

Step 1: Given information

We have given an integral $$\begin{gathered} \int {\int }_{R}xydA \\ Where R=\left\{\right(x,y\left)\right|0\le x\le 2,1\le y\le 4\} \end{gathered}$$

Step 2: Using iterated integral

Using the definition of iterated integrals we get,

$$\begin{gathered} {\int }_1^4{\int }_0^{2}xydxdy \\ ={\int }_1^4{{\left[\frac{x^{2}y}{2}\right]}^2}_{0}dy \\ ={\int }_1^{4}2ydy \\ ={{\left[y^2\right]}^4}_1=16-1 \\ =15 \end{gathered}$$