Area and Arc Length in Polar Coordinates

The polar area formula

Area between polar curves

Polar arc length

Numerical polar lengths

The area swept by r = f of theta from alpha to beta equals which integral?

Applying the polar area formula to the constant curve r = a over [0, 2 pi] gives which area?

The area between polar curves r_outer and r_inner over [alpha, beta] uses which integrand?

The polar arc length integrand is which expression?

To compute the area inside the cardioid r = 1 + cos theta but outside the circle r = 1, what must be found first?

One petal of r = cos 2 theta has area pi over 8. As a fraction of pi, what is it? Answer like pi/8.

The cardioid r = 1 + cos theta has perimeter 8. As a number, what is it? Answer with a number.

For the circle r = a constant, the arc length integrand simplifies because dr/d theta vanishes. What does the integrand become?

The infinitesimal polar area element is half r squared d theta. As a two-word name, what shape is it? Answer like small sector.

In the arc length integrand, the term (dr/d theta)^2 represents the contribution of which motion? Answer with two words.