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Cogito

AP Calculus BC · Chapter 2 · Lesson 2

Practice Worksheet

Area in Polar Coordinates

Sectors instead of rectangles.

12 problems · about 22 minutes · CHA-5.D

NameDate

Figure — use these to answer the problems

A sector is a fraction of a circle. Polar area sums infinitely many thin sectors.
d = 8
  • Diameter8 cm
  • Circumference25.13 cm
  • Circumference ÷ diameter3.14
Radius4
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. The circle r = 7 encloses an area of kπ. What is k?

    Answer

  2. Where does the one half in the polar area formula come from?

    • The area of a circular sector is one half r squared theta.
    • It is a convention with no reason.

  3. For r = 6, what is one half r squared?

    Answer

  4. The circle r = 3 encloses kπ. What is k?

    Answer

  5. What fraction sits in front of r squared in the polar area formula?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. The limits of a polar area integral are what? 1 angles, 2 x values.

    Answer

  2. The circle r = 5 encloses kπ. What is k?

    Answer

  3. For an area between curves, you subtract what? 1 the r values, 2 the r squared values.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the polar area setup in order.

    Write 1 to 4 in the boxes to put these in order.

    • Identify the region and the curve bounding it.
    • Find the angles where the region begins and ends.
    • Write the integral of one half r squared d theta.
    • Evaluate between those angles.

  2. For r = 2, what is one half r squared?

    Answer

  3. The Integrand

    For r = 4, what is one half r squared?

    Answer

  4. The Area

    The circle r = 4 encloses an area of kπ. What is k?

    Answer