Q. 41 - Calculus

Question

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

$$\begin{gathered} \int {\int }_R\sin (x+2y)dA \\ Where R=\left\{\right(x,y\left)\right|0\le x\le \pi ,0\le y\le \frac{\pi }{2}\} \end{gathered}$$

Step-by-Step Solution

Answer

The value of the integral is 0

Step 1: Given information

We are given that $$\begin{gathered} \int {\int }_R\sin (x+2y)dA \\ Where R=\left\{\right(x,y\left)\right|0\le x\le \pi ,0\le y\le \frac{\pi }{2}\} \end{gathered}$$

Step 2: Use iterated integral

We have,

$$\begin{gathered} {\int }_0^{\frac{\pi }{2}}{\int }_0^{\pi }\sin (x+2y)dxdy \\ ={\int }_0^{\frac{\pi }{2}}{\int }_0^{\pi }(\sin x\cos 2y+\sin 2y\cos x)dxdy \\ ={\int }_0^{\frac{\pi }{2}}{{[-\cos x\cos 2y+\sin 2y\sin x]}^{\pi }}_{0}dy \\ ={\int }_0^{\frac{\pi }{2}}\left(\cos 2y\right)dy \\ ={{\left[\frac{\sin 2y}{2}\right]}^{\frac{\pi }{2}}}_0 \\ =0 \end{gathered}$$