Problem 61 - Essential Calculus

Question

Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.

Step-by-Step Solution

Answer

The formula for the length of a polar curve is \[L=\int_a^b\sqrt{(dr/d\theta)^2+r^2}d\theta\].

Step 1: Definition of Polar Coordinates

First, remember the relation between polar and Cartesian coordinates. A point in the plane can be represented in polar coordinates as \((r,\theta)\) where \(r = \sqrt{x^2+y^2}\) and \(\theta = tan^{-1}(y/x)\). In terms of \(r\) and \(θ\), the Cartesian coordinates are expressed as \(x = r cos(θ)\) and \(y = r sin(θ)\).

Step 2: Parametric Form Conversion

Write these relationships in parametric form, with \(\theta\) as the parameter: \(x = f(\theta) = r(\theta) cos(\theta)\), \(y = g(\theta) = r(\theta) sin(\theta)\).

Step 3: Arc Length Formula and Derivatives

We know the arc length formula for parametric curves is defined as \[L=\int_a^b\sqrt{(dx/d\theta)^2 + (dy/d\theta)^2}d\theta\]. To use this formula to find the arc length of a polar curve, compute the derivatives \(dx/d\theta\) and \(dy/d\theta\). Using the chain rule, we get \(dx/d\theta=dr/d\theta cos(\theta)-r sin(\theta)\) and \(dy/d\theta=dr/d\theta sin(\theta)+r cos(\theta)\).

Step 4: Substitution

Substitute these derivatives back into the arc length formula: \[L=\int_a^b\sqrt{[(dr/d\theta cos(\theta)-r sin(\theta))^2+(dr/d\theta sin(\theta)+r cos(\theta))^2]}d\theta\].

Step 5: Expansion and Simplification

Expand and simplify the expression inside the square root: \[L=\int_a^b\sqrt{(dr/d\theta)^2+(r^2)}d\theta\].

Step 6: Ending Point

Finally, we have derived the formula for the length of a polar curve: \[L=\int_a^b\sqrt{(dr/d\theta)^2+(r^2)}d\theta\].