Understanding how to determine the area of an elliptical sector is an important concept in geometry. This step-by-step guide will walk you through the process using the integral formula, explanations, and real number examples to make it easy to follow and understand.
Step 1: Show the Formula
To calculate the area of an elliptical sector when the angle is given in degrees, we use the following integral formula:
\[ A = \int_{\theta_1}^{\theta_2} \frac{a^2 \cdot b^2}{2 \cdot (b^2 \cdot \cos^2(\theta) + a^2 \cdot \sin^2(\theta))} d\theta \]
where:
- \( A \) is the area of the sector.
- \( \theta \) is the angle parameter in degrees.
- \( a \) is the semi-major axis of the ellipse.
- \( b \) is the semi-minor axis of the ellipse.
Step 2: Explain the Formula
The integral formula \( A = \int_{\theta_1}^{\theta_2} \frac{a^2 \cdot b^2}{2 \cdot (b^2 \cdot \cos^2(\theta) + a^2 \cdot \sin^2(\theta))} d\theta \) calculates the area of the elliptical sector by integrating over the angle range \([\theta_1, \theta_2]\) in degrees. This approach accounts for the elliptical shape and varying radii at different angles.
Step 3: Use Actual Numbers as an Example
Let's assume we have an ellipse with a semi-major axis (\( a \)) of 6 units, a semi-minor axis (\( b \)) of 4 units, and we want to find the area of the sector from \( \theta_1 = 0^\circ \) to \( \theta_2 = 60^\circ \).
Step 4: Set Up the Integral
Substitute the given values into the formula:
\[ A = \int_{0^\circ}^{60^\circ} \frac{6^2 \cdot 4^2}{2 \cdot (4^2 \cdot \cos^2(\theta) + 6^2 \cdot \sin^2(\theta))} d\theta \]
\[ A = \int_{0^\circ}^{60^\circ} \frac{36 \cdot 16}{2 \cdot (16 \cdot \cos^2(\theta) + 36 \cdot \sin^2(\theta))} d\theta \]
\[ A = \int_{0^\circ}^{60^\circ} \frac{576}{2 \cdot (16 \cdot \cos^2(\theta) + 36 \cdot \sin^2(\theta))} d\theta \]
\[ A = \int_{0^\circ}^{60^\circ} \frac{288}{16 \cdot \cos^2(\theta) + 36 \cdot \sin^2(\theta)} d\theta \]
Step 5: Calculate the Integral
To solve the integral, we perform the integration:
\[ A \approx \int_{0^\circ}^{60^\circ} \frac{288}{16 \cos^2(\theta) + 36 \sin^2(\theta)} d\theta \]
Using numerical methods or a calculator:
\[ A \approx 14.3374 \, \text{square units} \]
Final Value
The area of the elliptical sector with a semi-major axis of 6 units, a semi-minor axis of 4 units, and a central angle from \(0^\circ\) to \(60^\circ\) is approximately \( 14.3374 \, \text{square units} \).
