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

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:

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 =

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 =

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

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

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).

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:

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 0 to π.

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.

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