Problem 76 - Essential Calculus

Question

Find the area of the region. $$ \begin{array}{l} x=2 \cot \theta \\ y=2 \sin ^{2} \theta \\ 0<\theta<\pi \end{array} $$

Step-by-Step Solution

Answer

The area of the region is \(2\pi\) square units.

Step 1: Express \(r\) in terms of \( \theta \)

From the given polar coordinates, you can express \(r\) where \(r^2 = x^2 + y^2\). In this case, \(r = \sqrt{(2 \cot \theta)^2 + (2 \sin^2 \theta)^2}\) which simplifies to \(r = 2\).

Step 2: Set up the integral

Now that you have found \( r = 2 \), you can set up the integral for the area of the region. This will be \(\int \frac{1}{2} r^2 d\theta\) over the region defined by \(0 < \theta < \pi\). So, the integral will be \(\int_0^\pi (2)^2/2 \, d\theta\).

Step 3: Integrate

Perform the integration of constant with respect to \(\theta\), it leads to \((2)^2/2 *\theta \Big|_0^\pi\).

Step 4: Apply limits

Applying the upper and lower limits of \(\pi\) and 0 to the result, you get \(2\pi\).

Key Concepts

Understanding Polar Coordinates

Understanding polar coordinates can be a bit tricky if you are used to the Cartesian coordinate system. Instead of using traditional x and y coordinates, polar coordinates use r (the radius) and θ (the angle). This system is particularly useful for problems involving circular or rotational symmetry.

In polar coordinates, a point is represented as (r, θ), where:

• r is the distance from the origin (0,0) to the point.
• θ is the angle measured from the positive x-axis to the line connecting the origin to the point.

The conversion between polar coordinates \( (r, \theta) \) and Cartesian coordinates (x, y) can be done using the formulas:

• \( x = r \cdot \cos\theta \)
• \( y = r \cdot \sin\theta \)

Thus, for a given curve in polar coordinates, the area can be found using these conversions when needed.

The Role of Integration in Area Calculation

Integration plays a crucial part in calculating the area under a curve or the area of a region defined by a parametric curve or polar equation. The process involves setting up an integral that covers the range of angles or the interval specified for the variable. In this exercise, the bounds for \( \theta \) are from 0 to \( \pi \).

For polar coordinates, the formula for calculating area uses the integral form:

• \( A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta \)

This formula accounts for the changes in angle and radius as they shape the curve being studied. By setting up the integral with the correct bounds and integrating, you find the total area enclosed by the curve.

Correctly applying the limits of integration is pivotal. In our solved exercise, substituting the limits of 0 and \( \pi \) into the evaluated integral results in calculating the area effectively.

The Importance of Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are essential in the study of parametric and polar equations. They help describe the way a curve behaves as it traces out its path.

In the context of polar coordinates, trigonometric functions convert the radius and angle into traditional x and y coordinates. For instance, in the exercise, \( x = 2 \cot \theta \) and \( y = 2 \sin^2 \theta \) employ the cotangent and sine squared functions.

These functions have specific characteristics that make them suitable:

• Sin and cosine functions are periodic and well-suited to describe rotating or oscillating systems.
• Cotangent is related to the tangent function and describes relationships in right triangles where the opposite side is known.

Understanding these functions and their properties allows one to effectively solve complex polar equations in many mathematical and physical problems.