Problem 15 - Thomas Calculus

Question

Use the transformation \(x=u / v, y=u v\) to evaluate the integral sum $$\int_{1}^{2} \int_{1 / y}^{y}\left(x^{2}+y^{2}\right) d x d y+\int_{2}^{4} \int_{y / 4}^{4 / y}\left(x^{2}+y^{2}\right) d x d y$$

Step-by-Step Solution

Answer

Apply transformation to simplify integration, then compute using Jacobian. Sum results.

Step 1: Understand the Given Integrals

The given problem asks us to evaluate the integral sum: \[ \int_{1}^{2} \int_{1 / y}^{y}(x^{2}+y^{2}) \, d x \, d y + \int_{2}^{4} \int_{y / 4}^{4 / y}(x^{2}+y^{2}) \, d x \, d y \]. This means we have two separate integrals over different regions. The challenge here is to simplify these integrals using a transformation.

Step 2: Apply the Transformation

The transformation given is \( x = \frac{u}{v} \) and \( y = u v \). We will apply this to the given integral by mapping the variables \((x, y)\) to \((u, v)\). Calculate the Jacobian of the transformation to adjust for the change of variables.

Step 3: Compute the Jacobian

Calculate the Jacobian of the transformation \((x = u/v, y = uv)\). The Jacobian \( J \) is given by: \[J = \begin{vmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix}\]Compute the partial derivatives, \( \frac{\partial x}{\partial u} = \frac{1}{v} \), \( \frac{\partial x}{\partial v} = -\frac{u}{v^2} \), \( \frac{\partial y}{\partial u} = v \), \( \frac{\partial y}{\partial v} = u \), and plug them into the determinant:\[J = \begin{vmatrix} \frac{1}{v} & -\frac{u}{v^2} \ v & u \end{vmatrix} = \frac{u}{v} + \frac{uv}{v^2} = \frac{u}{v} + \frac{u}{v} = \frac{2u}{v}\]

Step 4: Transform the Limits of Integration

Next, we transform the limits of integration using the transformation. For the first integral, we have \( 1 \leq y \leq 2 \) and \( \frac{1}{y} \leq x \leq y \). Transform these in terms of \(u\) and \(v\). Similarly, transform the limits for the second integral.

Step 5: Evaluate the Transformed Integrals

With the transformed variables and limits, the integral \( (x^2 + y^2) \, dx \, dy \) transforms to another expression in terms of \(u\), \(v\) and the Jacobian. Compute the transformed integrals step by step. This may involve polynomial integration and substitution using the new limits derived from the transformation.

Step 6: Sum the Evaluated Integrals

After evaluating both integrals in the transformed domain, sum the results to find the complete integral value as requested by the problem.

Key Concepts

Change of Variables

When dealing with complicated integrals, a technique called "change of variables" can simplify the process. This involves substituting the original variables with new ones through a specific transformation. In this particular exercise, the transformation is given as \( x = \frac{u}{v} \) and \( y = uv \).
This change of variables helps us express the integral in terms of new variables \( (u, v) \), potentially making the mathematical process easier.
It allows us to reframe the original function \((x^2 + y^2)\) in terms of these new variables, which often aligns the integration with more familiar forms or symmetries.
The ultimate goal of change of variables is often to simplify the function, the domain of integration, or both, to make the integral easier to solve.

Jacobian Determinant

The Jacobian determinant is a vital part of the change of variables technique. It's a factor that accounts for how areas or volumes scale under the transformation of coordinates. In simple terms, the Jacobian adjusts for the 'stretching' of space.
In our exercise, the Jacobian determinant for the transformation \( (x = \frac{u}{v}, y = uv) \) was computed as \( J = \frac{2u}{v} \).
This determinant ensures that when we substitute \( dx \) and \( dy \) with \( du \) and \( dv \), the new differential elements reflect the transformation correctly.
Here’s how you compute it: find the partial derivatives of the transformations and form a matrix, then calculate the determinant of that matrix.

• Partial derivatives \( \frac{\partial x}{\partial u} = \frac{1}{v} \), \( \frac{\partial x}{\partial v} = -\frac{u}{v^2} \).
• Partial derivatives \( \frac{\partial y}{\partial u} = v \), \( \frac{\partial y}{\partial v} = u \).

The determinant calculation is crucial for the correct evaluation of the integral.

Integration Limits

Transforming the integration limits is a crucial step when performing a change of variables. It involves determining the new boundary conditions that correspond to the original integration domain, thereby allowing us to correctly evaluate the integral in the new coordinate system.
In our exercise, we start with initial limits in \((x, y)\)-coordinates, such as \(1 \leq y \leq 2\) and \(\frac{1}{y} \leq x \leq y\), and find their counterparts in \((u, v)\)-coordinates.
This transformation often involves algebraic manipulation using the given relations \(x = \frac{u}{v}\) and \(y = uv\), and solving for \(u\) and \(v\) in terms of these limits.
Properly transforming the integration limits ensures that the region of integration remains accurate in the new variable system, aligning with the transformed integrand.

Coordinate Transformation

A coordinate transformation involves switching from one set of coordinates to another, usually to simplify calculations. In this problem, the transformation\( (x = \frac{u}{v}, y = uv) \) was utilized.
The transformation can provide a new perspective on the problem, exploiting symmetrical properties or easier-to-handle integrals.
By choosing an appropriate transformation, often based on the problem's structure or given relationships, we aim to rewrite expressions and boundaries in simpler terms.

• The idea is to make complex integrations less cumbersome.
• Coordinate transformations rely heavily on understanding the problem's geometric nature and choosing variables that simplify significant parts of the integrand or limits.

The result is an integral that's often much more straightforward to evaluate in the new coordinate space, supporting both rigorous and intuitive solutions.