Problem 49 - Essential Calculus

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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 \(\theta\).