Question

Give the integral formulas for area and arc length in polar coordinates.

Short answer

The area of a polar curve \( r = f(θ) \) is given by the integral formula \( A = \frac{1}{2}\int_{α}^{β} [f(θ)]^2 dθ \). The arc length of the same polar curve is given by the integral formula \( s = \int_{α}^{β} \sqrt{[f(θ)]^2 + [f'(θ)]^2} dθ \).

Step-by-step solution

State the Area Integral Formula in Polar Coordinates

The formula for the area \( A \) of a polar curve \( r = f(θ) \), for \( θ \) between alpha (α) and beta (β), is given by: \[ A = \frac{1}{2}\int_ {α}^{β} [f(θ)]^2 dθ \] This formula determines the area bound by the curve and the rays from the origin making angles α and β with the positive x-axis.

State the Arc Length Integral Formula in Polar Coordinates

The formula for the arc length \( s \) of a polar curve \( r = f(θ) \), for \( θ \) between alpha (α) and beta (β), is given by: \[ s = \int_{α}^{β} \sqrt{[f(θ)]^2 + [f'(θ)]^2} dθ \] This formula calculates the length of the curve traced out as \( θ \) varies from α to β.