Problem 28 - 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. $$\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$$

Step-by-Step Solution

Answer

Answer: The area of the ellipse is \(3\pi\sqrt{5}\).

Step 1: Identify the values of a and b

Compare the given ellipse equation \(\frac{x^{2}}{9}+\frac{y^{2}}{5}=1\) with the general ellipse equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). We can see that \(a^{2}=9\) and \(b^{2}=5\). Therefore, \(a=3\) and \(b=\sqrt{5}\).

Step 2: Calculate the area of the ellipse

Now that we know the values of \(a\) and \(b\), we can determine the area of the ellipse using the area formula \(A=\pi ab\). Substitute the values we found in the previous step: \(A = \pi (3)(\sqrt{5})\). Multiply to get the area of the ellipse: \(A = 3\pi\sqrt{5}\). So, the area of the given ellipse is \(3\pi\sqrt{5}\).

Key Concepts

Calculus

Calculus is a powerful branch of mathematics that deals with change and motion through the concepts of derivatives and integrals. It provides methods for solving problems involving complex shapes and motions where algebra and geometry alone may fall short. In the context of calculating the area of an ellipse, calculus allows us to approach the problem by considering an ellipse as a sum of infinitely many infinitesimally thin rectangles. By integrating over one of the ellipse's axes, calculus provides the means to sum up these tiny areas to find the total area enclosed by the ellipse.

In the case of the given exercise, the area calculation of an elliptical shape leverages the definite integral which computes the accumulation of quantities, like areas under a curve. The fundamental theorem of calculus bridges the concept of the antiderivative and the area under a curve, allowing us to use the area formula derived for an ellipse to determine its total area without the need for direct integration in this instance.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. This family includes shapes such as circles, ellipses, parabolas, and hyperbolas. An ellipse, being one of these conic sections, is the set of points where the sum of the distances from two fixed points, known as the foci, is constant.

In terms of its equation, an ellipse can be recognized by its canonical form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. The values of \(a\) and \(b\) are crucial as they not only define the shape and the size of the ellipse but also are directly used in the computation of the ellipse’s area. Understanding the nature of conic sections is fundamental when it comes to analyzing their properties and providing solutions to related problems, such as in the exercise provided.

Geometric Formulas

Geometric formulas are expressions that relate different geometrical attributes such as lengths, areas, and volumes. For an ellipse, the area formula \(A = \pi ab\) is derived from the general formula for the area of a circle, modifying it to account for the stretched radius along the x and y axes. In this equation, \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis.

Mastering these formulas allows students to solve a wide range of geometrical problems without having to re-derive the relationships from first principles every time. For the given exercise, by identifying the values of \(a\) and \(b\), students are applying their understanding of the geometric formula for the area of an ellipse. Knowing how to manipulate and substitute into these formulas is integral to finding solutions to textbook exercises and to gaining a deeper understanding of the geometry involved.