The Basic Idea
Finding the area between two polar curves is similar to finding the area between two rectangular curves, but there is one important difference: polar area is built from sectors, not vertical rectangles.
Suppose two curves are described by
For a particular angle , imagine the region stretching from the inner curve out to the outer curve. The area of a thin polar sector is determined by the square of its radius.
Therefore, the area between the curves is found by subtracting the inner sector from the outer sector and integrating with respect to .
This formula is the polar-coordinate version of the familiar "outer minus inner" idea from area between curves.
The Polar Area Formula
The area enclosed by a single polar curve from to is:
So if one curve lies outside another over the interval, the area between them is simply:
The factor appears in both terms. This is why the area between two polar curves involves the difference of the squares of the radii, not merely the difference of the radii.
The Most Important Step: Find the Intersections
Before setting up an area integral, you usually need to determine where the two curves intersect. If
then the intersection angles satisfy
Those intersection angles often become the limits of integration.
However, be careful: polar curves can represent the same point in more than one way. You may need to check the interval carefully rather than assuming that solving one equation gives every relevant intersection.
A Reliable Step-by-Step Method
- Write both polar equations clearly. Identify which radius belongs to each curve.
- Find the relevant intersection angles. Set the radii equal and solve on the interval that traces the desired region.
- Determine which curve is outer. Compare the values of the two radii at an angle inside the interval.
- Square the radii. Remember that polar area uses , not simply .
- Set up outer-minus-inner.
- Integrate over the correct angular interval.
- Check the result. The answer should be positive and should make geometric sense.
Worked Examples
Find the area of the region inside and outside .
Find the area inside the circle and outside the cardioid .
Suppose two curves are
and the desired region is traced over an interval where the curves intersect more than once.
How to Tell Which Curve Is Outside
This is one of the most common sources of errors.
For polar curves, "outside" means farther from the pole. It does not necessarily mean "higher" on the graph, because polar graphs are organized by angle rather than by vertical position.
Suppose the curves are and . Pick an angle between the intersection angles and evaluate both radii.
If , then is the outer curve at that angle.
A single test angle is often enough when you already know the curves only intersect at the endpoints of the interval. If they intersect again inside the interval, split the integral at that additional intersection.
Why the Radii Are Squared
Students sometimes write when they mean the area between two polar curves. That is incorrect.
The area of a thin sector is approximately
So for two curves, the outer sector contributes while the inner sector contributes . Subtracting gives:
That is the geometric reason the formula contains squared radii.
Using Symmetry
Symmetry can make a polar-area problem dramatically shorter.
If the region is symmetric about the polar axis, you can often find half the area and multiply by 2:
Similarly, if a region has rotational symmetry, you may be able to integrate over a smaller angular interval and multiply by the appropriate symmetry factor.
But do not use symmetry merely because the equations "look symmetric." Make sure the region being requested actually has the symmetry you are using.
Common Mistakes
The correct structure is , not . Polar area comes from , so the radius must be squared.
"Outer" means farther from the pole at a given angle. Test a convenient angle between the intersections instead of guessing from the appearance of the equations.
Polar curves can trace points in unusual ways. Make sure the interval actually traces the region requested by the problem. An equation may have additional solutions that belong to another part of the graph.
If the outer and inner curves switch roles, one integral with a single subtraction order will not work. Find the switching angle and split the area into separate integrals.
The polar area formula is . Leaving out the doubles the answer.
A Fast AP Calculus Strategy
When you see a free-response question asking for the area between two polar curves, use this checklist:
- Find the intersections. Set the two radii equal.
- Sketch or visualize the region. Identify exactly which part of the curves is being requested.
- Determine the outer curve. Test a point if necessary.
- Write outer squared minus inner squared.
- Multiply by .
- Use the intersection angles as limits when they bound the desired region.
- Split the integral if the curves change order.
- Check that the final answer is positive.
Practice Problems
Find the area inside and outside over the region between their consecutive intersections in the first quadrant.
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Find the area inside and outside for .
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The curves and intersect at several angles. Explain how you would determine which curve is outer on each interval before calculating the area.
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Set up, but do not evaluate, the area of the region inside and outside between their intersections.
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