How to calculate the area in polar coordinates?

To calculate the area in polar coordinates, we use the formula A = 1/2 ∫(r^2) dθ.

In polar coordinates, a point is represented by an angle θ and a distance r from the origin. To find the area enclosed by a polar curve, we need to integrate the function r^2 with respect to θ.

First, we need to find the limits of integration for θ. This can be done by finding the values of θ where the curve starts and ends. For example, if we want to find the area enclosed by the curve r = 2 + 4cosθ, we can see that the curve starts at θ = 0 and ends at θ = 2π.

Next, we need to find the function r^2. For the curve r = 2 + 4cosθ, we can square both sides to get r^2 = 4 + 16cosθ + 16cos^2θ.

Now we can substitute r^2 and the limits of integration into the formula A = 1/2 ∫(r^2) dθ.

A = 1/2 ∫(4 + 16cosθ + 16cos^2θ) dθ from θ = 0 to θ = 2π

We can simplify the integral by using the trigonometric identity cos^2θ = (1 + cos2θ)/2.

A = 1/2 ∫(4 + 16cosθ + 8 + 8cos2θ) dθ from θ = 0 to θ = 2π

A = 1/2 ∫(12 + 16cosθ + 8cos2θ) dθ from θ = 0 to θ = 2π

We can now integrate each term separately.

A = 1/2 [12θ + 16sinθ + 4sin2θ] from θ = 0 to θ = 2π

A = 1/2 [12(2π) + 16sin(2π) + 4sin(4π) - 12(0) - 16sin(0) - 4sin(0)]

A = 1/2 [24π]

A = 12π

Therefore, the area enclosed by the curve r = 2 +