Q. 6. - Calculus

Question

Explain how we arrive at the definite integral formula $\frac{1}{2}{\int }_{\alpha }^{\beta }(f{\left(\theta \right))}^2$in Theorem 9.13 for computing the area bounded by a polar function$r=f\left(\theta \right)$ on an interval $\left[\alpha ,\beta \right]$(Your explanation should include a limit of Riemann sums.) What would the integral ${\int }_{\alpha }^{\beta }f\left(\theta \right) d\theta$ represent?

Step-by-Step Solution

Answer

The answer is the sum of rectangles to the curve at each interval.

Step 1: Given information and finding the area of a region can be approximated by using sectors of circles.

Consider a polar function is $r=f\left(\theta \right)$that is defined in the interval $\theta =\alpha$and $\theta =\beta$

The area is estimated using the Riemann sum of approximation method.

Divide the region into sectors, and the total size of those sectors equals the region's area.

For a $k$th sector $S_k$is $\frac{1}{2}{\left(f\left({\theta }_k^{*}\right)\right)}^2\Delta \theta$

Step 2: Find the integral that represents the sum of the rectangles to the curve at each interval.

The region's approximate area $R$ is equal to the sum of the sectors' areas:

$R=\sum _{k=1}^n\frac{1}{2}{\left(f\left({\theta }_k^{*}\right)\right)}^{2}d\theta$

$\frac{1}{2}\left(f\right(\theta ){)}^2$this is the Riemann sum for the function on the interval $[\alpha ,\beta ]$

The region's approximate area $R$ is equal to the sum of the sectors' areas:

$n\to \infty$

The total of the rectangles to the curve at each interval is represented by the integral ${\int }_a^{b}f\left(x\right)d\theta$