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Find the exact length of the polar curve. r = θ2, 0 ≤ θ ≤ π?
Solution:
Given, r = θ2, 0 ≤ θ ≤ π
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 = \(\int_{0}^{\frac{5\pi }{4}}\sqrt{((\theta ^{2})^{2}+(2\theta )^{2})d\theta }\)
= \(\int_{0}^{\pi }\sqrt{(\theta^{4}+4\theta^{2})d\theta }\)
Taking out common term,
= \(\int_{0}^{\pi}\sqrt{\theta ^{2}(\theta^{2}+4)d\theta }\)
= \(\int_{0}^{\pi}\theta \sqrt{(\theta^{2}+4)d\theta }\)
= \(\left [ \frac{(\theta ^{2}+4)^{\frac{3}{2}}}{3}\right ]_{0}^{\pi}\)
= \(\frac{1}{3}\left [ (\pi)^{2}+4)^{\frac{3}{2}}-(0+4)^{\frac{3}{2}} \right ]\)
= \(\frac{1}{3}\left [ (\pi^{2}+4)^{\frac{3}{2}}-(4)^{\frac{3}{2}} \right ]\)
= \(\frac{1}{3}\left [ (\pi^{2}+4)^{\frac{3}{2}}-8\right ]\)
Therefore, the exact length of the polar curve is \(\frac{1}{3}\left [ (\pi^{2}+4)^{\frac{3}{2}}-8\right]\).
Find the exact length of the polar curve. r = θ2, 0 ≤ θ ≤ π?
Summary:
The exact length of the polar curve r = θ2, 0 ≤ θ ≤ π is \(\frac{1}{3}\left [ (\pi^{2}+4)^{\frac{3}{2}}-8\right ]\).
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