What is the formula for area in polar coordinates?

The formula for area in polar coordinates is given by A = ½ ∫[a,b] r² dθ.

To find the area enclosed by a polar curve, we need to integrate the function r² with respect to θ over the interval [a,b]. The formula for area in polar coordinates is A = ½ ∫[a,b] r² dθ. This formula is derived from the formula for the area of a sector of a circle, which is ½r²θ. In polar coordinates, the angle θ is replaced by the variable of integration, and the radius r is a function of θ.

To use this formula, we first need to find the limits of integration, which are the values of θ that correspond to the starting and ending points of the curve. We can then evaluate the integral using standard integration techniques. It is important to note that the function r² must be non-negative over the interval of integration, otherwise the formula will not give a valid result.

For example, let's find the area enclosed by the polar curve r = 2 + 4cos(θ) over the interval [0,2π]. First, we need to square the function to get r² = 4 + 16cos(θ) + 16cos²(θ). Then, we can substitute these values into the formula for area in polar coordinates:

A = ½ ∫[0,2π] (4 + 16cos(θ) + 16cos²(θ)) dθ
= ½ [4θ + 16sin(θ) + 8sin(2θ)]|[0,2π]
= ½ (8π)

Therefore, the area enclosed by the polar curve r = 2 + 4cos(θ) over the interval [0,2π] is 4π.