Question

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

Short answer

The integral formula for area in Polar Coordinates is \[\int_{a}^{b} \frac{1}{2} {r(\theta)}^{2} \, d\theta\] and for arc length is \[\int_{a}^{b} \sqrt{(r(\theta))^2 + \left(\frac{dr}{d\theta}\right)^2} \,d\theta\].

Step-by-step solution

Formulate the Integral Formula for Area in Polar Coordinates

To find the area between two radii, the integral formula in polar coordinates is: \[\int_{a}^{b} \frac{1}{2} {r(\theta)}^{2} \,d\theta\] where \(a\) and \(b\) are the limits of integration, \(r(\theta)\) is the polar function, and \(\theta\) is the angle.

Formulate the Integral Formula for Arc Length in Polar Coordinates

To find the arc length, the integral formula in polar coordinates is: \[\int_{a}^{b} \sqrt{(r(\theta))^2 + \left(\frac{dr}{d\theta}\right)^2} \,d\theta\] where \(a\) and \(b\) are the limits of integration, \(r(\theta)\) is the polar function and \(\frac{dr}{d\theta}\) is its derivative with respect to \(\theta\).