Double Integrals in Polar Coordinates

Round regions want round coordinates.

12 problems · about 24 minutes · F-IF.B.6, G-GPE.A.1

NameDate

Figure — use these to answer the problems

These evenly spaced rings are the level curves of a cone. In polar coordinates its height is just r.

a, the steepness1

Figure 1

Straightforward practice. Get the method working first.

5 problems

A disc of radius 7. What is its area divided by π?
Answer
2
Why does an extra factor of r appear in the polar area element?
- A polar wedge grows wider the further it sits from the origin.
- It is an arbitrary convention.
In polar coordinates, x² + y² becomes what power of r?
Answer
The polar area element is r to what power, times dr dθ?
Answer
A full circle. What is the upper θ limit, using 2π as about 6.28?
Answer

The same ideas with more to keep track of.

3 problems

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

4 problems

Sort each region by which coordinate system suits it better.
Write each item under the heading it belongs to: A disc of radius 4 · A square with sides on the axes · A ring between two circles · A rectangle 3 by 7
Polar coordinates
Rectangular coordinates
A point at r = 5 and θ = 0. What is its x-coordinate?
Answer
The Round Table
A disc of radius 5. What is its area divided by π?
Answer
The Half Disc
A half-disc of radius 4. What is its area divided by π?
Answer