A circle described in x and y needs awkward square-root limits. In polar coordinates it becomes r from 0 to R, which is as simple as limits get.
Step 1: Let's Learn
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The conversion
x = r cos θ and y = r sin θ, so x² + y² becomes simply r².
The extra r
The area element is not dr dθ but r dr dθ. Forgetting the r is the most common mistake in the whole chapter.
Why the r is there
A polar rectangle is a curved wedge, and wedges far from the origin are wider. The r accounts for that growing width.
Circular limits
A full disc of radius R has r from 0 to R and θ from 0 to 2π. A half-disc simply halves the θ range.
When to switch
Switch whenever the region is a disc, a ring or a wedge, or whenever x² + y² appears in the integrand.
Round regions want round coordinates
A disc has awkward Cartesian limits involving square roots and trivial polar limits. Changing coordinates to match the geometry of the region is usually the largest simplification available.
The extra r
The area element becomes r dr dθ, not dr dθ. The r accounts for sectors being wider further from the origin, and omitting it is the defining error in polar double integrals.
Setting polar limits
θ sweeps the angular extent of the region and r runs from the inner to the outer boundary at each angle. Sketching first is what makes both ranges obvious.
The classic payoff
The Gaussian integral, which has no elementary antiderivative in one variable, is evaluated by squaring it and converting to polar coordinates. It is the standard demonstration that the change of variables is powerful, not merely convenient.
Step 2: Try It Yourself
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Evenly spaced rings mean the steepness never changes as you climb.
Step 3: Watch an Example
One step at a time.
Watch Zoya Find a Disc Area
Zoya computes the area of a disc of radius 3 using polar coordinates.
- Step 1
She integrates the constant 1 with the area element r dr dθ.
Step 4: Your Turn
Practice makes it stick.
The Round Table
Problem 1 of 2
A disc of radius 5. What is its area divided by π?
The Half Disc
Problem 2 of 2
A half-disc of radius 4. What is its area divided by π?
Go Round
1 of 8
In polar coordinates, x² + y² becomes what power of r?
2 of 8
The polar area element is r to what power, times dr dθ?
3 of 8
A full circle. What is the upper θ limit, using 2π as about 6.28?
4 of 8
A disc of radius 6. What is its area divided by π?
5 of 8
A quarter disc of radius 2. What is its area divided by π?
6 of 8
A ring between radius 3 and radius 5. What is its area divided by π?
7 of 8
Sort each region by which coordinate system suits it better.
Tap something to move it.
- Empty
- Empty
8 of 8
A point at r = 5 and θ = 0. What is its x-coordinate?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A disc of radius 7. What is its area divided by π?
Question 2 of 2
Why does an extra factor of r appear in the polar area element?
What You Learned
- 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.