 In this video, we'll build intuition for how we calculate areas enclosed by polar curves. The idea is very similar to what we use to interpret an integral of a function over an interval as the area under the curve in that interval. Recall that we used Riemann's sums to approximate the area under that curve. We'll use a similar idea, this time with areas of sectors of circles. Let's remind ourselves of how we calculated areas of sectors of circles in geometry. So here we have a circle. We know its area is pi R squared. And the entire angle measure is 360 degrees, or as we know it, 2 pi. Well, suppose we look at just a portion of this circle, a portion whose angle measure is theta. We know then that the area of that sector, just this piece, the area of this sector, must be a portion or a fraction of the area of the entire circle. Since the angle measure is theta, we know that the fraction of the entire circle that this sector represents is theta over the 2 pi, which is the entire angle measure. Therefore, the area of this sector is theta over 2 pi times pi R squared. So the area of this sector is therefore theta over 2 times R squared. Now, if we're looking at a polar curve, let's say it's shaped this way, just as we did before with areas under curves in the development of the integral, suppose we take this section of the polar curve and divide it up into little mini sectors of circles. And just as we did with Riemann sums to approximate the area under a curve, we'll find the area of each of these sectors and add them up. So let's consider the i-th sector. We'll call this the i-th sector. And we're going to define these sectors with a theta of size delta theta. So I've got a number of sectors, each of whose angles are delta theta. And as we said in the previous screen, we can find the area of each of these sectors by taking that theta, in our case delta theta, over 2 times the radius associated with that sector squared. This is the area of the i-th sector. Well, once we find the areas of each of these sectors and I add them up, I should get some sort of approximation for the area of the whole polar curve. So the idea is I'm adding all of these up. And then in order to get a better and better approximation, I'm really looking at what this area is approaching as delta theta gets smaller and smaller, which means I'm really looking at smaller and smaller segments in order to better approximate the area in this polar curve. So ultimately, I'm interested in letting delta theta approach 0. And I'm adding up the areas of each of these sectors, which we now know as an integral. We're evaluating it from our initial theta to the theta of interest. So theta 1, we'll call it to theta 2. 1 half r squared d theta. So this is the area of a polar curve r of theta between theta 1 and theta 2.