9.9 Finding the Area of the Region Bounded by Two Polar Curves

When working with polar functions, finding the area between two curves is a conceptually similar to finding the area between two curves in Cartesian coordinates. The main difference is that instead of subtracting the y-coordinates, you'll be subtracting the radii ("outer minus inner"). 🎯

To find the area between two polar curves, you can use the same process as finding the area under one curve. First, you'll need to determine the interval of integration for each of the two curves. Once you have determined the intervals, you'll integrate the outer curve with respect to θ, and then integrate the inner curve with respect to θ. Subtracting the definite integral of the inner curve from the definite integral of the outer curve will give you the area between the two curves.

For example, consider two polar functions: r1(θ) and r2(θ). To find the area between the two curves, you can use the following formula:

It's important to keep in mind that the area between two curves is a signed quantity. This means that it can be positive or negative depending on the order of the curves. It is positive if the outer curve is above the inner curve and negative if the opposite is true.

Additionally, it's crucial to be mindful of the endpoints of integration while finding the area between two curves, if the two curves intersect, the area between them will be the sum of the area of the regions enclosed by the two curves between the point of intersection.

Keep in mind that the curves may intersect, which can make finding the area between them more complex. In cases where the curves intersect, you may need to find the area inside the curves by splitting the region into multiple sections.

For example, consider two polar functions, r1(θ) and r2(θ), where r1(θ) represents the outer curve and r2(θ) represents the inner curve. If the two curves intersect at a certain point, you'll need to find the area between the two curves by splitting the region into two sections: the area enclosed by the outer curve and the area enclosed by the inner curve.

You can find the area inside the curves by integrating each curve separately, then subtracting the area enclosed by the inner curve from the area enclosed by the outer curve.

Example: Let R be the region inside the graph of the polar curve r = 4 and r = 4 + 2sin(2θ) on [0, π]. Find the area of R.

Since the graphs intersect at θ = 2 we can see that when θ < π/2, r = 4 is on top, and when θ > π/2, r = 4+2sin(2θ) is on top. Based on this information, we can construct two integrals:

There is one last thing you need to know about polar functions: arc length. Finding arc length is pretty straightforward, but you do need to have the formula memorized for the exam.