Answer:
Length of polar arc =
[tex] \displaystyle\frac{1}{3}\bigg[ \bigg(\frac{9\pi^2}{4} + 4\bigg)^\frac{3}{2} - 8 \bigg][/tex]
Step-by-step explanation:
We have to find the arc length of the polar curve.
We are given the following:
[tex]r(\theta) = f(\theta) = \theta^2,\\\\0 \leq \theta \leq \displaystyle\frac{3\pi}{2}[/tex]
Length of polar curve =
[tex]\displaystyle\int_b^a \sqrt{r^2 + \bigg(\displaystyle\frac{dr}{d\theta}\bigg)^2}~d\theta[/tex]
Putting the values:
[tex]\displaystyle\frac{dr}{d\theta} = \frac{d(\theta^2)}{d \theta} = 2\theta\\\\\text{Length of polar curve} = \displaystyle\int_b^a \sqrt{r^2 + \bigg(\displaystyle\frac{dr}{d\theta}\bigg)^2}~d\theta\\\\= \displaystyle\int^\frac{3\pi}{2}_0 \sqrt{\theta^4 + (2\theta)^2}~d\theta\\\\= \displaystyle\int^\frac{3\pi}{2}_0 \sqrt{\theta^4 + 4\theta^2}~d\theta\\\\= \displaystyle\int^\frac{3\pi}{2}_0 \theta\sqrt{\theta^2+4}~d\theta \\\\= \bigg|\frac{(\theta^2 + 4)^\frac{3}{2}}{3}\bigg|_0^\frac{3\pi}{2}[/tex]
[tex]= \displaystyle\frac{1}{3}\bigg[ \bigg(\frac{9\pi^2}{4} + 4\bigg)^\frac{3}{2} - (4)^\frac{3}{2}\bigg]\\\\= \displaystyle\frac{1}{3}\bigg[ \bigg(\frac{9\pi^2}{4} + 4\bigg)^\frac{3}{2} - 8 \bigg][/tex]
