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Chapter 5: Problem 4

Calculate the area enclosed by ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (Figure)

Short Answer

Expert verified
The area enclosed by the ellipse is \( \pi a b \).

Step by step solution

01

Understand the Formula of an Ellipse

The general formula for an ellipse is given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a \) is the semi-major axis, and \( b \) is the semi-minor axis. To find the area of an ellipse, you will use these axes values in a specific formula for area.
02

Area Formula for an Ellipse

The area \( A \) of an ellipse is calculated using the formula \( A = \pi a b \). This formula uses the lengths of the semi-major and semi-minor axes directly.
03

Plug in the Values of the Axes

Since the ellipse is given in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the values for the semi-major and semi-minor axes are already represented as \( a \) and \( b \) respectively. Therefore, substitute these values into the area formula.
04

Calculate the Area

Substitute \( a \) and \( b \) into the formula \( A = \pi a b \). Therefore, the area is \( A = \pi a b \). This represents the total area enclosed by the ellipse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ellipse Formula
The equation of an ellipse is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This formula represents a standard form of an ellipse centered at the origin on the coordinate plane. The variables \(x\) and \(y\) represent any point located on the ellipse, while \(a\) and \(b\) are lengths related to the axes of the ellipse.
  • Standard Form: The ellipse's formula is rearranged to equal 1, ensuring it remains standardized.
  • Axes Representation: The denominator \(a^2\) is associated with the x-axis, and \(b^2\) with the y-axis.
Understanding the ellipse formula is crucial as it provides a straightforward method to identify key dimensions of the ellipse. By analyzing the formula, you can determine which axis is longer and thereby deduce whether the ellipse is oriented horizontally or vertically.
Semi-Major Axis
The semi-major axis of an ellipse is denoted as \(a\). It represents half the length of the longest diameter of the ellipse. The term "major" conveys that this axis is the largest when comparing it to the semi-minor axis.
  • Definition: It is the longer radius that extends from the center out towards the edge of the ellipse.
  • Positioning: The semi-major axis can be aligned with either the x-axis or the y-axis.
For an ellipse aligned horizontally, \(a\) is positioned on the x-axis. If the ellipse is aligned vertically, \(a\) is on the y-axis. This distinction is vital when determining the orientation of the ellipse, providing crucial information for interpreting its geometric nature.
Semi-Minor Axis
Complementarily, the semi-minor axis, denoted by \(b\), refers to half the length of the shortest diameter of the ellipse. It is shorter than the semi-major axis, as suggested by the term "minor."
  • Definition: It is the shorter radius that similarly extends from the center toward the edge of the ellipse.
  • Placement: Consequently, when \(a\) is on the x-axis, \(b\) will be on the y-axis, and vice versa.
Together with the semi-major axis, the semi-minor axis helps define the exact shape and size of the ellipse. Understanding the role of each axis can assist in various calculations related to the ellipse, such as determining its area or modeling elliptical shapes in practical scenarios.

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Most popular questions from this chapter

[T] Evaluate \(\iint_{S} x^{2} z d S\), where \(S\) is the portion of cone \(z^{2}=x^{2}+y^{2}\) that lies between planes \(z=1\) and \(z=4\).

Find the mass of a lamina of density \(\xi(x, y, z)=z\) in the shape of hemisphere \(z=\left(a^{2}-x^{2}-y^{2}\right)^{1 / 2}\).

Compute \(\iint_{S} \mathbf{F} \cdot \mathbf{N} d S\), where \(\mathbf{F}(x, y, z)=2 y z \mathbf{i}+\left(\tan ^{-1} x z\right) \mathbf{j}+e^{x y} \mathbf{k}\) and \(\mathbf{N}\) is an outward normal vector \(S\), where S is the surface of sphere \(x^{2}+y^{2}+z^{2}=1\).

Evaluate the following integrals.\(\int_{C} x^{2} d y+(2 x-3 x y) d x\), along \(C: y=\frac{1}{2} x\) from \((0,0)\) to \((4,2)\)

For the following exercises, describe each vector field by drawing some of its vectors. $$ \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} $$
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