Polar Coordinates

Question

I dont really know how to go about this question. I know that the area is

$$\int_\alpha^\beta \frac12 r^2\, d\theta $$

The question is to find the area of the shaded region.

Answer 1

If you read through the links I left on the comments, you should see that the area that's trapped between two polar curves should be:

$$\int_{\alpha}^\beta \frac12 r^2_{out} d\theta - \int_{\alpha}^\beta \frac12 r^2_{in} d\theta$$

Or more compactly,

$$\frac12 \int_{\alpha}^\beta (r^2_{out} - r^2_{in}) d\theta.$$

Note: This formula should remind you of the formula you learned for finding area between two functions.

$r_{out}$ corresponds to the curve that is further out, further away from the origin. (Remember $r$ represents the distance from the origin.)

$r_{in}$ represents the curve that is closer to the origin, closer to the origin.

$\alpha$ and $\beta$ represent the angle (in polar coordinates) of the intersection points of the two curves. To find $\alpha$ and $\beta$, you need to set the two curves in question and solve for $\theta$.

Let me know if you need more help. I can update this post with more details if needed.