Find the area of the region that lies inside the cardioid r=1+cosθand outside the circle r=1.

Name: Problem 28, Chapter 14: University Calculus: Early Transcendentals
Uploaded: 2020-04-29T03:45:53-08:00
Duration: 6 min 1 s
Channel: Bobby Barnes
Description: Find the area of the region that lies inside the cardioid r=1+cosθand outside the circle r=1.

Find the area of the region that lies inside the cardioid...

Key Concepts

Area between Polar Curves

When dealing with regions bounded by two polar curves, the area can be found by calculating the difference between the areas of the larger and smaller regions. This typically involves integrating the square of the outer curve and subtracting the integral of the square of the inner curve over the same interval of angle.

Intersection of Polar Curves

Determining where two polar curves intersect involves setting their equations equal and solving for the angle ?. These intersection points are crucial for establishing the limits of integration when calculating areas that involve more than one curve, such as regions defined by the overlap or difference of two curves.

Polar Coordinates

Polar coordinates represent points in a plane by a distance from a fixed point (the pole) and an angle from a fixed direction. This system simplifies the description and analysis of curves that are naturally expressed in terms of angles and radii, such as circles and cardioids.

Area in Polar Coordinates

The formula for finding the area enclosed by a polar curve r = f(?) is (1/2)?[? to ?] (f(?))^2 d?. This method inherently accounts for the way area is distributed in sectors of a circle, making it especially useful for regions bounded by curves defined in terms of the radius as a function of the angle.

Transcript

00:01 We want to find the area of the region that lies inside the cardioid r is equal to 1 plus cosine theta and outside the circle r is equal to 1.

00:10 So i'm going to head and graph to both of those.

00:12 So outside of the circle and inside our cardioid would actually be this region right here that i'm going to shade in red.

00:25 So within this region, we just have to figure out how our radius varies as well as theta.

00:32 Radius will we start from the origin and we go until we intersect our region.

00:36 So the lower bound is going to be one.

00:39 And then we keep on going until we pass through the region, which is going to be our cardioid.

00:46 So r is going to vary 1 up until 1 plus cosine of theta.

00:55 Now the next thing we need to do is figure out how theta varies.

00:59 Well, notice that in this region we would go from here, rotate all the way around, and then exit through here.

01:07 So at least looking at this, it's pretty easy to see that this is going to be negative pi -half to pi -half for our rotation, since it's just the axes.

01:19 But if we weren't given that, what we could do is set these two equations equal to each other and then solve.

01:24 So the algebraic way to do this would just be to let 1 equal to 1 plus cosine theta.

01:31 That gives us 0 is equal to cosine theta.

01:33 And well, we know that would be negative pi half and pi half, or at least for two of the solutions.

01:39 So we have those.

01:42 Now we can go ahead and plug everything into the equation that they give us.

01:48 And that's going to be, so we want to put theta last, since our radius is going to depend on it.

01:56 So it's going to go from negative pi half to pi half.

01:59 And then our radius goes from one to one plus cosine of theta.

02:04 R, dr, d, theta.

02:06 So we can go ahead and integrate this.

02:09 And that's going to give 1 .5r squared.

02:12 So i'm just going to factor that 1 half all the way out.

02:14 So it would be 1 .5 negative pi half to pi half of r squared evaluated from 1 plus cosine theta, or 1 to 1 plus cosine theta.

02:29 Then we have d theta on the outside...

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