Question

how can I Calculate the area of the figures presented in the polar coordinate system by means of the definite integrals:

Answer 1

Answer:

The area is

`(\pi)/(2)`

Step-by-step explanation:

The area is given as:

`\begin{gathered} A=(1)/(2)\int ^{(\pi)/(2)}_0(2\sin 3\varphi)^2d\varphi \\ \\ =(1)/(2)\int ^{(\pi)/(2)}_04\sin ^23\varphi d\varphi \\ \\ =(4)/(2)\int ^{(\pi)/(2)}_00.5(1-\cos 3\varphi)d\varphi \\ \\ =\int ^{(\pi)/(2)}_0(1-\cos 3\varphi)d\varphi \\ \\ =(\varphi-(\sin3\varphi)/(3))\begin{cases}(\pi)/(2) \\ 0\end{cases} \\ \\ =((\pi)/(2)-(\sin(3\pi)/(2))/(3))-(0-(\sin 0)/(3)) \\ \\ =(\pi)/(2) \end{gathered}`