Question

In exercise 3, set up and evaluate a double integral that represents the area of the region.

Answer 1

The final answer will be $12\pi$ .

We will be making use of polar coordinates in order to evaluate the above integral. That is, rather than x and y, we will make use of and $\theta$ where,

$x=r\cos{\theta}$

and

$y=r\sin{\theta}$.

Range of r will be from 2 to 4, as clearly seen from the diagram. And range of $\theta$ will be from to $2\pi$ , since all the quadrants are colored.

We will set up the untequal uning cosuduates to make it sumplen to polar solve.. clucle r=2 and neue, auca Cuiside the lies outside the circle r=4 . The augue a goes the from oo to 360° T since colored region goes full circle]. Now, ditequal becomes : for drdo lou & is Jacobian coordinates an 5274 da 22 16 -4 7 l 72 12 do P = 12 / da da - 12 za = 12 2 Tol L - 12T