Problem 20

Question

Use a transformation to evaluate the given double integral over the region $R$ which is the triangle with vertices $(1,0),(4,0)$, and $(4,3) .$ $$ \iint_{R}(2 x-y) \cos (y-2 x) d A $$

Step-by-Step Solution

Verified

Answer

Use the transformation $u = 2x - y, v = y - 2x$ and calculate the integral with new limits.

1Step 1: Identify the Region in the xy-plane

The region $R$ is the triangle with vertices $(1,0), (4,0), (4,3)$ in the xy-plane. To identify the transformation, observe the triangle's shape: it is a right triangle with a vertical side along $x=4$ and a base along the x-axis from $x=1$ to $x=4$.

2Step 2: Establish a Transformation

To simplify the integration, introduce new variables. Let $u = 2x - y$ and $v = y - 2x$. These variables help change the complex integral into one that might be easier to integrate. Solve for $x$ and $y$ in terms of $u$ and $v$ to find the Jacobian.

3Step 3: Find the Jacobian of the Transformation

Compute the partial derivatives: $\frac{\partial x}{\partial u}, \frac{\partial x}{\partial v}, \frac{\partial y}{\partial u}, \frac{\partial y}{\partial v}$. Use these to form the Jacobian determinant $J$, which is given by: \[J = \begin{vmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix} \].

4Step 4: Set Up New Integral with Limits

Find the new limits of integration for $u$ and $v$ based on the transformation and the original region $R$. Substitute $x$ and $y$ in terms of $u$ and $v$ into the integrand $(2x-y)\cos(y-2x)$, and multiply by the absolute value of the Jacobian determinant $|J|$.

5Step 5: Evaluate the Transformed Integral

Integrate over $u$ and $v$ using the new limits. Calculate the innermost integral, followed by the outer integral. Simplify the expression to find the final result. Make sure to verify all calculations and variable substitutions are correctly applied.

Key Concepts

Jacobian determinantTransformationLimits of integration

Jacobian determinant

When performing transformations for double integrals, the Jacobian determinant is a critical component. It essentially measures how a transformation changes the area in the plane.
For a transformation involving two new variables, say $ u $ and $ v $, you express the original variables $ x $ and $ y $ as functions of $ u $ and $ v $. The Jacobian determinant $ J $ is computed using:

Partial derivatives: these describe how each original variable changes with respect to the new variables.
Formula: \[J = \begin{vmatrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\end{vmatrix}\]

Evaluating this determinant will give you the factor by which area elements change under the transformation. This factor is crucial to correctly adjusting the limits of integration and scaling the integrand in the new coordinate system. A positive Jacobian signifies that the orientation of the region is preserved, while a negative Jacobian indicates a reversal.

Transformation

A transformation is an essential strategy in solving double integrals, especially when dealing with complex regions. It involves a change of variables aimed at simplifying the integrand or the region of integration.
In the given problem, the transformation uses new variables defined as:

$ u = 2x - y $
$ v = y - 2x $

These new variables can potentially turn a complicated region into one that's easier to handle. Here’s how you go about it:

Express the original variables $ x $ and $ y $ in terms of $ u $ and $ v $.
Determine how these substitutions transform the original region.
Identify the relationships and ensure all parts of the integral accurately reflect the new variables.

This transformation aims to change the original complex triangle in the $ xy $-plane into a simpler region in the $ uv $-plane, often a rectangle. The ease of integration over a regular shape often outweighs the complexity of setting up the transformation.

Limits of integration

Gradient of setting new limits is crucial when switching from the original variables $ x $ and $ y $ to new ones, like $ u $ and $ v $. This comes after determining the transformation equations.
For setting up new integration limits using the transformation:

First, sketch or understand the transformed region. Use the relations between $ u $ and $ v $ and the vertices of the triangle.
Assess where each side of the original region transfers in the new coordinate system.
Establish the minimum and maximum values for $ u $ and $ v $. These dictate the new limits.

In our initial problem, the transformation morphs the triangular region in the $ xy $-plane into a rectangle or simpler shape. With your new limits of integration verified, you can now proceed with transforming the integral. This makes solving the integral substantially easier and more straightforward.

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