9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

3 min read•june 18, 2024

Sumi Vora

Jed Quiaoit

AP Calculus AB/BC ♾️

279 resources

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Simple Area under Polar Curves

When we calculate the area under the curve for Cartesian graphs, we use integration to find the area between the curve and the x-axis. This is typically done by dividing the area into a series of rectangles and approximating the area by summing the areas of the rectangles. 📏

However, when we calculate the area under the curve for polar functions, we use a different approach. A polar function is defined on a circular plane, and the area under the curve is calculated by summing the areas of small triangles that make up the curve.

Imagine a pizza being cut into thin slices. Each slice represents an infinitely thin triangle that makes up the curve. The area of each triangle is calculated using the formula A = (1/2)bh, where h is the radius (r) and b is the base (r * dθ) of the triangle.

The reason we use the formula A = (1/2)bh is that the area of each triangle is proportional to the radius, multiplied by the tiny value d to obtain an infinitely tiny base. By summing the areas of all the triangles, we can find the total area under the curve, which gives us important information about the function such as the total distance traveled by a moving object, the total charge density in an electric field, or the total flux in a magnetic field.

In addition, we can use polar coordinates to calculate the area enclosed between two polar curves, by subtracting the area under one curve from the area under the other. This can be useful in physics and engineering to calculate the enclosed area of a region and in engineering to calculate the area of a surface! 🧑‍✈️

When calculating the area under a curve for polar functions, one of the most important steps is determining the interval on which you want to integrate. This can sometimes be challenging, especially if you are not given the graph of the function or if you are working on a non-calculator section of an exam.

However, there are a few general guidelines you can follow to help you determine the interval:

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 typically be from 0 to 2π. This is because these functions go around a full circle once within that interval, and the graph will repeat itself every 2π units.
If the equation has a coefficient inside the trig function (such as sin4θ or cos2θ), then the function likely has multiple "petals" or loops, and the interval will not be from 0 to 2π. In this case, you can make a chart to find where the function meets zero, and then 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, and it's recommended to double-check the solution.

Additionally, the polar functions may contain a term r^n, this term will affect the interval of the integration, the interval of integration will be where the function is defined i.e. r > 0 or r > a, etc.