Problem 12 - Thomas Calculus

Question

The area of an ellipse the area \(\pi a b\) of the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) can be found by integrating the function \(f(x, y)=1\) over the region bounded by the ellipse in the \(x y\) -plane. Evaluating the integral directly requires a trigonometric substitution. An easier way to evaluate the integral is to use the transformation \(x=a u, y=b v\) and evaluate the transformed integral over the disk \(G : u^{2}+v^{2} \leq 1\) in the \(u v\) -plane. Find the area this way.

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We are given an ellipse with the equation \( x^{2}/a^{2} + y^{2}/b^{2} = 1 \) and asked to find its area using integration after transforming to a new coordinate system. We will use the transformation \( x = au \) and \( y = bv \), which will transform the region to a disk \( G: u^{2} + v^{2} \leq 1 \) in the \( uv \)-plane. The aim is to find the area within this disk, which corresponds to the area of the ellipse in the original \( xy \)-plane.

Step 2: Perform Coordinate Transformation

Using the transformation \( x = au \) and \( y = bv \), we substitute these into the ellipse equation: \( (au)^{2}/a^{2} + (bv)^{2}/b^{2} = 1 \), which simplifies to \( u^{2} + v^{2} \leq 1 \). The region is now a standard disk in the \( uv \)-plane.

Step 3: Calculate the Jacobian of Transformation

The Jacobian for the transformation from \( (x,y) \) to \( (u,v) \) is given by the determinant of the matrix of partial derivatives:\[J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \begin{vmatrix} a & 0 \ 0 & b \end{vmatrix} = ab\]Therefore, the Jacobian \( J = ab \), which needs to be included in the integral transformation.

Step 4: Convert and Evaluate the Integral

The area of the ellipse is given by the integral of \( f(x,y) = 1 \) over the ellipse region. Replacing with the transformation and Jacobian, the integral becomes:\[\text{Area} = \int_{u^{2}+v^{2} \leq 1} 1 \cdot ab \, dudv\]This simplifies to \( ab \times \text{Area of the disk} \). The area of the disk \( u^{2}+v^{2} \leq 1 \) is \( \pi \), thus the integral evaluates to \( ab \pi \).

Step 5: Conclusion

Using the coordinate transformation, the area of the ellipse \( x^{2}/a^{2}+y^{2}/b^{2}=1 \) is found to be \( ab \pi \), which corresponds to integrating over a transformed circular region with the factor from the Jacobian included. This confirms the formula for the area of an ellipse, \( \pi ab \).

Key Concepts

Coordinate Transformation

When we talk about coordinate transformation, we're simply changing the way we look at a certain problem. In this exercise, we start with the ellipse equation \( x^{2}/a^{2} + y^{2}/b^{2} = 1 \).We know that direct integration can be tricky, so we transform this ellipse into a shape that's easier to manage: a circle or disk in another plane, the \( uv \)-plane.
By using the transformations \( x = au \) and \( y = bv \), we simplify the equation into \( u^{2} + v^{2} \leq 1 \), which is the equation of a circle with radius 1.

• This makes our region a unit circle in the \( uv \)-plane.
• Transforming coordinates helps change complex shapes (like ellipses) into simpler ones (like circles), making integral calculations easier.

The main goal of a coordinate transformation is to redefine how areas and volumes are represented under a different mathematical lens.

Jacobian Determinant

The Jacobian determinant is crucial when performing coordinate transformations, especially for integration. It's essentially a scale factor that tells us how area, volume, or other measures change under transformation.In our exercise, when transforming from \( (x, y) \) coordinates to \( (u, v) \) coordinates, the Jacobian determinant helps us correct the integral for this change. We take partial derivatives of the transformations:

• \( x = au \), resulting in \( \frac{\partial x}{\partial u} = a \)
• \( y = bv \), resulting in \( \frac{\partial y}{\partial v} = b \).

The Jacobian matrix is \[\begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \begin{vmatrix} a & 0 \ 0 & b \end{vmatrix} = ab\]This determinant, \( ab \), is then used in the integral to ensure we're accurately calculating the area in the new \( uv \)-coordinate system. It's an essential tool to maintain the integrity of measurements when coordinates change.

Integral Evaluation

Evaluating integrals can be surprisingly straightforward if approached correctly. For finding the area of an ellipse, the integral calculates the sum of areas in tiny pieces across the space within the boundary.In the transformed \( uv \)-system, we evaluate the area as \[\int_{u^{2}+v^{2} \leq 1} 1 \cdot ab \, dudv \]

• The expression \( 1 \cdot ab \) indicates the area element is scaled by the Jacobian determinant \( ab \).
• \( dudv \) represents tiny elements of area within the \( uv \)-disk.

As the \( uv \)-disk is a simple circle with radius 1, the area is known to be \( \pi \). Thus, when multiplied by \( ab \), it gives the total area of the ellipse:\[\text{Area} = ab \times \pi = ab \pi\]This reveals not only the elegance of integral calculus but also the power of coordinate transformations in simplifying complex problems.

Trigonometric Substitution

Trigonometric substitution is a method that can simplify integrals, particularly those involving square roots. However, in the context of finding an ellipse's area, it's used less directly.Originally, if we were to stay in the \( xy \)-plane, we might use trigonometric identities to handle the shapes bounded by curves. For an ellipse, this involves substitution like \( x = a \cos \theta \) and \( y = b \sin \theta \), but these can complicate integration.In our solution, instead, we favor a direct geometrical transformation to a disk, sidestepping the cumbersome trigonometric functions altogether.

• While trigonometric substitutions are essential in integration techniques, by transforming to the \( uv \)-plane, we manage the integral more directly.
• This approach prevents unnecessary complexities that these substitutions usually handle.

So, while trigonometric substitution is a valuable tool, the coordinate transformation renders this problem more accessible.