Problem 4 - Essential Calculus

Question

Find the area of the region bounded by the graph of the polar equation using (a) a geometric formula and (b) integration. $$ r=3 \cos \theta $$

Step-by-Step Solution

Answer

The area of the region bounded by the polar equation r=3cos(\(\theta\)) is \(2.25\pi\) using geometric formula and \(4.5\pi\) using integration.

Step 1: Solving Using Geometrics

First, the formula for the area of a circle, \(A = \pi r^2\), should be used. The circle's radius is half of the maximum value of r, given by r=3cos(\(\theta\)), so the radius is r = 1.5. Therefore, the area A = \(\pi * (1.5)^2\).

Step 2: Solving Using Integration

Alternatively, another approach is using integration. The area of a polar graph is given by \(0.5* \int_{0}^{\pi} (3cos(\(\theta\)))^2 d \theta\). Integrating this equation gives the area under the curve defined by the polar equation from 0 to pi.

Step 3: Calculating the Integral

Evaluating the integral gives an alternate solution for the area. The integral works out to \(0.5* \int_{0}^{\pi} 9cos^2(\(\theta\))d \theta\), which can be solved to \(4.5\pi\).

Key Concepts

Area under a curve

When working with graphical representations, finding the area under a curve is a common problem, especially in calculus. This concept involves determining the space between the curve and the horizontal axis. In polar coordinates, curves are described using the radial distance from the origin and an angle. For instance, in the given exercise, we have the polar equation \(r = 3 \cos \theta\), which describes a specific curve.Finding the area under a curve involves mathematical techniques that depend on the shape of the curve:

• For simple geometrical shapes like circles or semi-circles, we use predefined geometric formulas based on symmetry and known dimensions.
• For more complex shapes, or precise areas beneath the polar curve, calculus techniques such as integration help calculate the exact area.

These methods allow us to understand and quantify the space occupied by different polar curves, essential for real-world applications ranging from physics to engineering.

Integration in polar coordinates

Integration in polar coordinates is a powerful technique to find the area of regions defined by polar equations. It's particularly useful when geometric formulas aren't directly applicable. The formula used is:\[A = \frac{1}{2} \int_{}^{} r^2 \, d\theta\]This formula helps calculate the area by integrating the square of the radial distance \(r\) over a defined range of angles \(\theta\). Let's break it down with our example:- The equation \(r = 3 \cos \theta\) describes a curve in polar coordinates. To find the area enclosed by this curve, we would integrate:\[A = \frac{1}{2} \int_{0}^{\pi} (3\cos \theta)^2 \, d\theta\]- This helps sum up small radial segments over the given range of \(\theta\), capturing the entire enclosed area.Integration in polar coordinates is quite intuitive once the relationship between \(r\) and \(\theta\) is understood. It allows for flexible area calculations beyond simple geometric interpretations.

Geometric formulas for circle areas

Geometric formulas offer straightforward ways to find areas of standard shapes like circles and sectors. For circles, the basic formula is:\[A = \pi r^2\]This formula gives the area by using the radius of the circle. In the original exercise, we identified that the polar equation \(r = 3 \cos \theta\) resembles a circle. However, \(r\) changes with \(\theta\), requiring us to determine an effective radius for the circle.Let's walk through the concept:

• First, observe that the maximum value of \(r\) is 3 when \(\theta = 0\), creating a full circle. However, due to symmetry, we consider half this maximum value, leading to an effective radius of 1.5 for a semi-circle.
• Using the geometric formula, the area becomes \(\pi \times (1.5)^2\).

Geometric formulas are efficient for standard shapes, providing a quick way to calculate areas when applicable, showcasing the elegance and utility of fundamental geometry principles.