Question

Find the area enclosed by the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\).

Short answer

The area of the ellipse is \( 6\pi \).

Step-by-step solution

Identify the Formula for Area of an Ellipse

The formula for the area of an ellipse is given by \( A = \pi \times a \times b \), where \( a \) and \( b \) are the semi-major and semi-minor axes of the ellipse, respectively.

Determine the Semi-Major and Semi-Minor Axis

In the equation of the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), the values under the variables represent squares of the lengths of the semi-axes. Therefore, \( a = 3 \) and \( b = 2 \), as \( 3^2 = 9 \) and \( 2^2 = 4 \).

Calculate the Area of the Ellipse

Substitute \( a = 3 \) and \( b = 2 \) into the area formula: \( A = \pi \times 3 \times 2 = 6\pi \).

Final Result for the Area

The area enclosed by the ellipse is \( 6\pi \). Thus, solving using the given dimensions completes the calculation.

Key concepts

Ellipse Equation

An ellipse is a beautiful geometric shape that looks like a stretched out circle. To describe an ellipse mathematically, we use the ellipse equation. The standard form of this equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This equation allows us to understand the shape and size of the ellipse by identifying the lengths of its axes.
In the equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), the numbers 4 and 9 reveal important information about the ellipse. They are the squares of the lengths of the axes that define the shape. So when you see these numbers, think about finding the square roots to identify the axes' actual lengths.
Understanding the ellipse equation helps us calculate the area, dimensions, and sometimes even the position of an ellipse on coordinate planes.

Semi-Major Axis

The semi-major axis is one of the most important parts of an ellipse. It represents half of the longest diameter of the ellipse. We often denote this dimension as \( a \).
In our example, the semi-major axis comes from the larger denominator in the ellipse equation. Here, \( b = 9 \) under \( y^2 \) is bigger than \( a = 4 \) under \( x^2 \). This implies that 9 is associated with our semi-major axis. Taking the square root, we find that \( a = 3 \).
Thus, the semi-major axis is the value that stretches the ellipse vertically in this case. Being aware of the major axis's significance helps in visualizing how broad or slender an ellipse is.

Semi-Minor Axis

The semi-minor axis is another essential part of the ellipse. It measures half of the shortest diameter of the ellipse, often denoted as \( b \).
In the ellipse equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), the value of 4 sits under \( x^2 \), which corresponds to our semi-minor axis. By taking the square root, we find that \( b = 2 \).
So in our example, the semi-minor axis is 2, indicating how broad the ellipse is along the horizontal direction. Knowing the length of the semi-minor axis completes the full picture of the ellipse's dimensions and assists in calculating properties like area.

Calculus

Calculus is a powerful mathematical tool often used to understand the behavior and properties of shapes and equations. When dealing with ellipses, calculus can help in more advanced topics, such as finding tangents or calculating volumes of revolution. However, the basic calculation of area does not require complex calculus but follows through a formula easily derived from geometric principles.
The area of an ellipse is given by the formula \( A = \pi \times a \times b \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. This straightforward formula comes from the integration techniques of calculus, but don’t worry! For basic area calculations, you only need this neat formula.
Remember, understanding calculus deepens insight into why formulas work, but the computation itself often remains simple with the right steps.