Find the exact length of the polar curve. r = θ, 0 ≤ θ ≤ π/2

Solution:

Given, r = θ, 0 ≤ θ ≤ π/2

We have to find the exact length of the polar curve.

The formula for calculating the Arc Length is given by

\(L=\int_{a}^{b}\sqrt{r^{2}+(\frac{dr}{d\theta })^{2}}d\theta\)

Here, r² = θ²

dr/dθ = 1

So, \(L=\int_{0}^{\frac{\pi }{2}}\sqrt{\theta ^{2}+(1)^{2}}d\theta\)

\(L=\int_{0}^{\frac{\pi }{2}}\sqrt{\theta ^{2}+1}d\theta\)

\(L=\left [\frac{\theta \sqrt{\theta ^{2}+1}}{2}+\frac{asinh(\theta )}{2} \right ]_{0}^{\frac{\pi }{2}}\)

\(L=\left [\frac{\frac{\pi }{2} \sqrt{(\frac{\pi }{2})^{2}+1}}{2}+\frac{asinh(\frac{\pi }{2})}{2}-0 \right ]\)

\(L=\left [\frac{\pi \sqrt{(\frac{\pi }{2})^{2}+1}}{2}+\frac{asinh(\frac{\pi }{2})}{2}\right]\)

Therefore, the exact length of the polar curve is \(L=\left [\frac{\pi \sqrt{(\frac{\pi }{2})^{2}+1}}{2}+\frac{asinh(\frac{\pi }{2})}{2}\right]\)

Find the exact length of the polar curve. r = θ, 0 ≤ θ ≤ π/2

Summary:

The exact length of the polar curve. r = θ, 0 ≤ θ ≤ π/2 is \(L=\left [\frac{\pi \sqrt{(\frac{\pi }{2})^{2}+1}}{2}+\frac{asinh(\frac{\pi }{2})}{2}\right]\).