Determine the area of the ellipse x²/4 + y²/9 = 1 using method of integration.

To find the area of the ellipse given by the equation:

\[
\frac{x^2}{4} + \frac{y^2}{9} = 1
\]

using integration, we will proceed step by step. The equation describes an ellipse with semi-major axis \(a = 3\) and semi-minor axis \(b = 2\), as it can be rewritten as:

\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \quad \text{where} \quad a = 3 \quad \text{and} \quad b = 2.
\]

### Step 1: Setup of the double integral
We want to compute the area of the ellipse, and we will do this by setting up a double integral in Cartesian coordinates. The total area \(A\) of the ellipse can be given by:

\[
A = 4 \int_0^a \int_0^{y_{\text{max}}(x)} dy \, dx
\]

where \(y_{\text{max}}(x)\) is the upper bound of the ellipse for a

Find the area of the ellipse x²/4 + y²/9 = 1 using integration methods. This detailed guide covers the ellipse equation, double integral setup, and step-by-step solution to compute the area using calculus techniques. Suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation