Cogito
Multivariable Calculus · Chapter 7 · Lesson 2
Double Integrals in Polar Coordinates
Round regions want round coordinates.
12 problems · about 24 minutes · F-IF.B.6, G-GPE.A.1
What this lesson teaches
The student converts double integrals to polar coordinates and applies the extra factor of r.
- Polar coordinates turn awkward circular limits into simple constant ones.
- x² + y² becomes r², which simplifies most circular integrands immediately.
- The area element is r dr dθ, and the extra r must never be dropped.
Warm Up
Straightforward practice. Get the method working first.
5 problemsA disc of radius 7. What is its area divided by π?
Answer 49
Why 49.
Why does an extra factor of r appear in the polar area element?
Answer A polar wedge grows wider the further it sits from the origin.
Why The wedges widen with radius.
In polar coordinates, x² + y² becomes what power of r?
Answer 2
Why r squared.
The polar area element is r to what power, times dr dθ?
Answer 1
Why A single factor of r.
A full circle. What is the upper θ limit, using 2π as about 6.28?
Answer 6.28
Why One full turn.
Build It Up
The same ideas with more to keep track of.
3 problemsA disc of radius 6. What is its area divided by π?
Answer 36
Why 6 squared.
A quarter disc of radius 2. What is its area divided by π?
Answer 1
Why A quarter of 4.
A ring between radius 3 and radius 5. What is its area divided by π?
Answer 16
Why 25 − 9.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each region by which coordinate system suits it better.
Answer Polar coordinates: A disc of radius 4, A ring between two circles · Rectangular coordinates: A square with sides on the axes, A rectangle 3 by 7
Why Round shapes want round coordinates.
A point at r = 5 and θ = 0. What is its x-coordinate?
Answer 5
Why cos 0 is 1.
The Round Table: A disc of radius 5. What is its area divided by π?
Answer 25
Why 25.
The Half Disc: A half-disc of radius 4. What is its area divided by π?
Answer 8
Why 8.