Q. 17 - Calculus

Question

Find a definite integral expression that represents the area of the given region in the polar plane and then find the exact value of expression

The region bounded enclosed by the spiral $r=\theta$ and the x-axis on the interval $0\le \theta \le \pi$

Step-by-Step Solution

Answer

The integral can be given as $\frac{1}{2}{\int }_0^{\pi }{\theta }^{2}d\theta$. The exact value of integral can be given as 5.165 units

Step 1: Given information

We are given an equation of spiral as $r=\theta$

Step 2: Evaluate

The area can be given as

$$\begin{gathered} A=\frac{1}{2}{\int }_0^{\pi }{r}^{2}d\theta \\ A=\frac{1}{2}{\int }_0^{\pi }{\theta }^{2}d\theta \\ A=\frac{1}{2}(10.33) \\ A=5.165unit \end{gathered}$$