2. Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve.
Consider the region bounded by the curve r = f (θ) and the two rays
To derive a formula for the area, divide the region into N narrow sectors of angle
corresponding to a partition of the interval

3. Recall that a circular sector of angle Δθ and radius r has
If Δθ is small, the jth narrow sector is nearly a circular sector of radius rj = f (θj), so its area is
The total area is approximated by the sum:

4. This is a Riemann sum for the integral
If f (θ) is continuous, then the sum approaches the integral as
and we obtain the following formula.
THEOREM 1 Area in Polar Coordinates If f (θ) is a continuous function, then the area bounded by a curve in polar form r = f (θ) and the rays θ =

5. We know that r = R defines a circle of radius R
We know that r = R defines a circle of radius R. By THM 1, the area is equal to
THEOREM 1 Area in Polar Coordinates If f (θ) is a continuous function, then the area bounded by a curve in polar form r = f (θ) and the rays θ =

6. Use Theorem 1 to compute the area of the right semicircle with equation r = 4 sin θ.
The equation r = 4 sin θ defines a circle of radius 2 tangent to the x-axis at the origin. The right semicircle is “swept out” as θ varies from 0 to
By THM 1, the area of the right semicircle is

7. Sketch r = sin 3θ and compute the area of one “petal.”
r varies from 0 to 1 and back to 0 as θ varies from 0 to
r varies from 0 to
-1 and back to 0 as θ varies from
r varies from 0 to 1 and back to 0 as θ varies from

8. The two circles intersect at the points where (r, 2 cos θ) = (r, 1) or
The area between two polar curves r = f1(θ) and r = f2(θ) with f2(θ) ≥ f1(θ), for
is equal to
Area Between Two Curves Find the area of the region inside the circle r = 2 cos θ but outside the circle r = 1.
The two circles intersect at the points where (r, 2 cos θ) = (r, 1) or
in other words, when 2 cos θ = 1.
Region (I) is the difference of regions (II) and (III).

9. We close this section by deriving a formula for arc length in polar coordinates. Observe that a polar curve r = f (θ) has a parametrization with θ as a parameter:

10. Find the total length of the circle r = 2a cos θ for a > 0.
f (θ) = 2a cos θ
Note that the upper limit of integration is π rather than 2π because the entire circle is traced out as θ varies from to π.

11. To find the slope of a polar curve r = f (θ), remember that the curve is in the x-y plane, and so the slope is
Since x = r cos θ and y = r sin θ, we use the chain rule.

12. Find an equation of the line tangent to the polar curve r = sin 2θ when