Cogito
AP Calculus BC · Chapter 2 · Lesson 2
Area in Polar Coordinates
Sectors instead of rectangles.
12 problems · about 22 minutes · CHA-5.D
What this lesson teaches
The student computes areas bounded by polar curves.
- Polar area integrates one half r² dθ.
- The limits are angles, found where the region begins and ends.
- For a region between curves, subtract the squared radii.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThe circle r = 7 encloses an area of kπ. What is k?
Answer 49
Why 49.
Where does the one half in the polar area formula come from?
Answer The area of a circular sector is one half r squared theta.
Why The sector area formula.
For r = 6, what is one half r squared?
Answer 18
Why 36 ÷ 2.
The circle r = 3 encloses kπ. What is k?
Answer 9
Why 3².
What fraction sits in front of r squared in the polar area formula?
Answer 0.5
Why From the sector area.
Build It Up
The same ideas with more to keep track of.
3 problemsThe limits of a polar area integral are what? 1 angles, 2 x values.
Answer 1
Why The variable is theta.
The circle r = 5 encloses kπ. What is k?
Answer 25
Why 5².
For an area between curves, you subtract what? 1 the r values, 2 the r squared values.
Answer 2
Why Square first, then subtract.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the polar area setup in order.
Answer 1. Identify the region and the curve bounding it. 2. Find the angles where the region begins and ends. 3. Write the integral of one half r squared d theta. 4. Evaluate between those angles.
Why The limits are found before the integral is written.
For r = 2, what is one half r squared?
Answer 2
Why 4 ÷ 2.
The Integrand: For r = 4, what is one half r squared?
Answer 8
Why 8.
The Area: The circle r = 4 encloses an area of kπ. What is k?
Answer 16
Why 16.