Question

Find the exact length of the polar curve. r = θ2, 0 ≤ θ ≤ 3π/2

Answer 1

Hello my friend it is the 21 and 31

Answer 2

Answer:

Length of polar arc =

$\displaystyle\frac{1}{3}\bigg[ \bigg(\frac{9\pi^2}{4} + 4\bigg)^\frac{3}{2} - 8 \bigg]$

Step-by-step explanation:

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

We are given the following:

$r(\theta) = f(\theta) = \theta^2,\\\\0 \leq \theta \leq \displaystyle\frac{3\pi}{2}$

Length of polar curve =

$\displaystyle\int_b^a \sqrt{r^2 + \bigg(\displaystyle\frac{dr}{d\theta}\bigg)^2}~d\theta$

Putting the values:

$\displaystyle\frac{dr}{d\theta} = \frac{d(\theta^2)}{d \theta} = 2\theta\\\\\text{Length of polar curve} = \displaystyle\int_b^a \sqrt{r^2 + \bigg(\displaystyle\frac{dr}{d\theta}\bigg)^2}~d\theta\\\\= \displaystyle\int^\frac{3\pi}{2}_0 \sqrt{\theta^4 + (2\theta)^2}~d\theta\\\\= \displaystyle\int^\frac{3\pi}{2}_0 \sqrt{\theta^4 + 4\theta^2}~d\theta\\\\= \displaystyle\int^\frac{3\pi}{2}_0 \theta\sqrt{\theta^2+4}~d\theta \\\\= \bigg|\frac{(\theta^2 + 4)^\frac{3}{2}}{3}\bigg|_0^\frac{3\pi}{2}$

$= \displaystyle\frac{1}{3}\bigg[ \bigg(\frac{9\pi^2}{4} + 4\bigg)^\frac{3}{2} - (4)^\frac{3}{2}\bigg]\\\\= \displaystyle\frac{1}{3}\bigg[ \bigg(\frac{9\pi^2}{4} + 4\bigg)^\frac{3}{2} - 8 \bigg]$