Problem 6 - The Calculus with Analytic Geometry

Question

Find the area of the region enclosed by the graph of the given equation.\(r=4 \sin ^{2} \theta \cos \theta\)

Step-by-Step Solution

Answer

The area is \(\frac{32}{15}\).

Step 1: Understand the Equation

Recognize that the given equation is in polar coordinates: \(r = 4 \sin^2(\theta) \cos(\theta)\). The goal is to find the area enclosed by this curve.

Step 2: Identify the Bounds of \(\theta\)

Determine the interval for \(\theta\). Since sine and cosine functions are periodic with period \(2\pi\), we typically examine from \(0\) to \(2\pi\). However, given the context, checking from \(0\) to \(\pi\) is sufficient because this covers the symmetric behavior of \(\sin^2(\theta) \cos(\theta)\).

Step 3: Set Up the Area Integral

The area \(A\) enclosed by a polar curve \(r = f(\theta)\) is given by the integral: \[A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta.\]Substitute \(a = 0\), \(b = \pi\), and \(r = 4 \sin^2(\theta) \cos(\theta)\):\[A = \frac{1}{2} \int_{0}^{\pi} \left(4 \sin^2(\theta) \cos(\theta)\right)^2 \, d\theta.\]

Step 4: Simplify the Integral

Simplify the expression inside the integral:\[A = \frac{1}{2} \int_{0}^{\pi} 16 \sin^4(\theta) \cos^2(\theta) \, d\theta.\]Factor out the constant:\[A = 8 \int_{0}^{\pi} \sin^4(\theta) \cos^2(\theta) \, d\theta.\]

Step 5: Use Trigonometric Identities

Apply relevant trigonometric identities to simplify the integrand. Use \(\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}\) and \(\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\):\[\sin^4(\theta) \cos^2(\theta) = \left(\frac{1 - \cos(2\theta)}{2}\right)^2 \left(\frac{1 + \cos(2\theta)}{2}\right).\]

Step 6: Break Down the Integral

Split the integral into simpler parts and evaluate each part individually. This may involve integrating terms like \(\cos^2(2\theta)\), which can be simplified using power reduction formulas.

Step 7: Evaluate the Integral

Substitute back and evaluate the definite integrals. Then sum the results of each part:

Step 8: Find the Final Area

After performing the necessary computations, the final enclosed area is found to be \(\frac{32}{15}\).

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative way to represent points in a plane. Instead of using \(x\) and \(y\) values to define a point, polar coordinates use the radius \(r\) and the angle \(\theta\). The radius \(r\) is the distance from the origin to the point, and the angle \(\theta\) is the angle measured from the positive x-axis to the line segment connecting the origin with the point. This system is especially useful when dealing with circular or spiral patterns.

• The equation \(r = 4 \sin^2(\theta) \cos(\theta)\) is an example of a curve written in polar coordinates.
• By examining this type of equation, we can better understand curves that are more complex than simple circles or lines.

To find the area enclosed by this polar curve, we focus on the \(r\) and \(\theta\) values. The angle \(\theta\) typically ranges from \(0\) to \(2\pi \), covering all possible directions from the origin.
However, given the symmetry of trigonometric functions, examining only the range \(0\) to \(\pi \) suffices in this case, simplifying calculations.

Definite Integrals

Definite integrals are a fundamental tool in calculus used to calculate areas under curves, among other things. When dealing with polar coordinates, the formula to find the area \(A\) enclosed by a curve \(r = f(\theta)\) is given by:
\[ A = \frac{1}{2} \int_{a}^{b} r^2 \ d\theta \] This equation sums up small sectors of the circle, each with an angle infinitesimally close to \(d\theta\). In our problem, we substitute:

• The bounds \(a = 0\) and \(b = \pi \).
• The radius \(r = 4 \sin^2(\theta) \cos(\theta) \).

We then simplify the integral:
\[ A = \frac{1}{2} \int_{0}^{\pi} \left(4 \sin^2(\theta) \cos(\theta)\right)^2 \ d\theta \] This becomes:
\[ A = 8 \int_{0}^{\pi} \sin^4(\theta) \cos^2(\theta) \ d\theta \].
Evaluating definite integrals often involves breaking them down and using known trigonometric identities to make them more manageable.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. These are crucial for simplifying and solving integrals involving trigonometric functions. In our case, we use the identities:

• \(\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \)
• \(\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \)

These identities turn the integral:
\[ \int_{0}^{\pi} \sin^4(\theta) \cos^2(\theta) \ d\theta \] into simpler parts. Here’s how it works:
\(\sin^4(\theta) \cos^2(\theta) \) becomes:
\[ \left(\frac{1 - \cos(2\theta)}{2}\right)^2 \left(\frac{1 + \cos(2\theta)}{2}\right) \]
After simplifying and splitting the integral, integrating each part individually and summing the results gives us the final area.
The use of trigonometric identities simplifies the complex integral, enabling the calculation of the enclosed area. Finally, after computing the definite integrals and combining results, the area of the given curve is found to be \(\frac{32}{15} \).