Q. 7 - Calculus

Question

Fill in the blanks to complete each of the following theorem statements: 

Let  $\alpha$ and $\beta$  be real numbers such that $0\le \beta -\alpha \le 2\mathrm{\pi }$.Let R be the region in the polar plane bounded by the rays $\theta =\alpha$ and $\theta =\beta$ and a positive continuous function$r=f\left(\theta \right)$. Then the area of region R is ................

Step-by-Step Solution

Answer

The blank is filled by $\frac{1}{2}{\int }_{\alpha }^{\beta }{\left(f\left(\theta \right)\right)}^{2}d\theta$.

Step 1. Given information

R is the region in the polar plane bounded by the rays $\theta =\alpha$ and $\theta =\beta$ and a positive continuous function $r=f\left(\theta \right)$.

Step 2. Area of the region.

Since the area of a sector of an circle having radius $r$ and center angle $\varphi$ rad is:

$A=\frac{1}{2}\varphi r^2$

The area of the region will be the limit of the Riemann sum as $n\to \infty$.

$\underset{n\to \infty }{\lim }\underset{1}{\overset{n}{\sum \frac{1}{2} }}{\left(f\left({\theta }_k\right)\right)}^2∆\theta$

So, 

 

The area of the region is;

$$\begin{gathered} A=\frac{1}{2}{\int }_{\alpha }^{\beta }{r}^{2}d\theta \\ =\frac{1}{2}{\int }_{\alpha }^{\beta }\left(f(\theta \right){)}^{2}d\theta \end{gathered}$$