Find the exact length of the polar curve. r = θ², 0 ≤ θ ≤ 2π

Solution:

Given, r = θ², 0 ≤ θ ≤ 2π

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

The length of the polar curve is given by \(\int_{a}^{b}\sqrt{(r^{2}+(\frac{dr}{d\theta })^{2})d\theta }\)

Now, \(\frac{dr}{d\theta }= \frac{d(\theta )^{2}}{d\theta }=2\theta\)

Length of the polar curve = \(\int_{0}^{2\pi}\sqrt{((\theta ^{2})^{2}+(2\theta )^{2})d\theta }\)

= \(\int_{0}^{2\pi }\sqrt{(\theta^{4}+4\theta^{2})d\theta }\)

Taking out common term,

= \(\int_{0}^{2\pi}\sqrt{\theta ^{2}(\theta^{2}+4)d\theta }\)

= \(\int_{0}^{2\pi}\theta \sqrt{(\theta^{2}+4)d\theta }\)

= \(\left [ \frac{(\theta ^{2}+4)^{\frac{3}{2}}}{3}\right ]_{0}^{2\pi}\)

= \(\frac{1}{3}\left [ (2\pi)^{2}+4)^{\frac{3}{2}}-(0+4)^{\frac{3}{2}} \right ]\)

= \(\frac{1}{3}\left [ (4\pi^{2}+4)^{\frac{3}{2}}-(4)^{\frac{3}{2}} \right ]\)

= \(\frac{1}{3}\left [ (4\pi^{2}+4)^{\frac{3}{2}}-8\right ]\)

Therefore, the length of the polar curve is \(\frac{1}{3}\left [ (4\pi^{2}+4)^{\frac{3}{2}}-8\right ]\).

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Find the exact length of the polar curve. r = θ², 0 ≤ θ ≤ 2π

Summary:

The exact length of the polar curve. r = θ², 0 ≤ θ ≤ 2π is \(\frac{1}{3}\left [ (4\pi^{2}+4)^{\frac{3}{2}}-8\right ]\).