When we calculate the area under the curve for Cartesian graphs, we would integrate with rectangles, since it is a rectangular plane. However, when we find the area under the curve for polar functions, we need to add up the area of triangles (imagine cutting a pizza into really really thin slices).
Since the area of a triangle is calculated by (1/2)bh, where h = r and b = rdθ (the base would be proportional to the radius, multiplied by the tiny value d to obtain an infinitely tiny base).
The tricky part about calculating the area is finding the interval on which you want to integrate. Sometimes, they will give you the graph of the function, or you will be able to graph it on your calculator. However, for non-calculator sections, you might have to figure out the endpoints just with the function.
Here are a few general guidelines you can follow.
If the equation has a sinθ or cosθ without any squares or coefficients, then the interval will always be from 0 to 2π, since it will go around a full circle once within that interval
If the equation has a coefficient inside the trig function (sin4θ or cos2θ, etc), then it the function likely has multiple petals. You can make a chart in order to find where the function meets zero, and then just calculate the area of one petal and multiply it by the number of petals.
If you can’t figure out the boundaries, you can usually guess that they are 0 to 2π. However, you should only use this as a last resort.
Example: Find the area of the region enclosed by the polar curve r=sin4
First, since there is a coefficient inside of the sine function, we can assume that there will be petals to the function
We can figure out the length of one petal by making a chart:
We can see that this pattern will continue; the graph will come back to the origin 8 times over [0, 2), so there are 8 petals