Problem 32 - Contemporary Precalculus

Question

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$5 x^{2}+y^{2}=5$$

Step-by-Step Solution

Answer

Answer: The area of the ellipse is \(\pi\sqrt{5}\).

Step 1: Rewrite the given ellipse equation in the standard form

Divide both sides of the equation by 5: $$\frac{5x^2}{5} + \frac{y^2}{5} = \frac{5}{5}$$ which simplifies to: $$\frac{x^2}{1} + \frac{y^2}{5} = 1$$

Step 2: Identify the values of a and b from the standard ellipse equation

The equation is in the form of \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). Comparing this with the given equation, we have \(a^2=1\) and \(b^2=5\). Hence, \(a=1\) and \(b=\sqrt{5}\).

Step 3: Calculate the area of the ellipse using the formula \(\pi\)ab

Plug in the values of a and b into the formula: $$\text{Area} = \pi \cdot a \cdot b = \pi \cdot 1 \cdot \sqrt{5} = \pi \sqrt{5}$$ The area of the ellipse is \(\pi\sqrt{5}\).

Key Concepts

Standard Form of Ellipse Equation

The concept of the standard form of an ellipse equation is crucial to understanding how to deal with ellipses in mathematics. An ellipse is a geometric shape that looks like a stretched-out circle. When dealing with the equation of an ellipse, it is important to rewrite it in its standard form to easily identify the important parameters of the ellipse.

The standard form of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This equation is a more general form, specifying that an ellipse is parallel to the x and y-axis, and offers parameters \(a\) and \(b\), denoting the semi-major and semi-minor axes respectively.

In the provided exercise, the original ellipse equation is \(5x^2 + y^2 = 5\). To convert it into the standard form, we divided the entire equation by 5. This process helps in identifying the values of \(a\) and \(b\). Once in the standard form, comparing coefficients \((x^2/1 + y^2/5 =1)\) helps us to determine that \(a^2 = 1\) and \(b^2 = 5\). These values are essential to further calculations.

Calculating Area in Calculus

Calculating the area of an ellipse involves a bit of calculus. Unlike a circle, which has a straightforward area formula, the area for an ellipse involves integration when initially derived. Thankfully, there's a convenient formula that can be used once you know the characteristics of the ellipse.

The area of an ellipse is given by the formula:\[\text{Area} = \pi \cdot a \cdot b\]where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. This formula arises by determining the area that an ellipse covers on a 2D plane.

In our exercise, once the ellipse equation was in standard form and we identified \(a = 1\) and \(b = \sqrt{5}\), we directly used the area formula:- Substitute \(a = 1\)- Substitute \(b = \sqrt{5}\)Resulting in the calculation:\(\text{Area} = \pi \cdot 1 \cdot \sqrt{5} = \pi \sqrt{5}\). Calculus aids us in understanding and deriving the formula, but application is quite straightforward once the parameters are known.

Ellipse Parameters

When exploring the geometry of an ellipse, understanding its parameters is key to many calculations, including finding its area. The main parameters of any ellipse include the lengths of its semi-major and semi-minor axes.

In the standard form of an ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\):

• \(a\): represents the length of the semi-major axis. For horizontal ellipses, \(a\) is the larger number. Vertical ellipses have the opposite demeanor, with \(b\) being larger.
• \(b\): stands for the length of the semi-minor axis. Conversely in terms of magnitude with \(a\).

In our given ellipse, once the equation \((x^2/1 + y^2/5 = 1)\) was simplified, we found:

• \(a = 1\)
• \(b = \sqrt{5}\)

These parameters allow you to predict and describe the orientation and size of the ellipse.

By knowing these values, calculations involving the ellipse such as the area or boundary points become straightforward tasks.