Math Problem Statement

Use the given transformation to evaluate the integral. The transformation is defined as u = y - x, v = y + x, and the integral is ∫∫_R e^(y - x)/(y + x) dx dy, where R is a trapezoid with vertices at (3, 0), (8, 0), (0, 3), (0, 8).

Solution

To solve this problem, let's break it down into steps.

Step 1: Identify the given transformation

The problem gives us a transformation: u=y−x,v=y+xu = y - x, \quad v = y + x

This is a transformation from the variables (x,y)(x, y) to (u,v)(u, v). We will need to find the Jacobian determinant of this transformation in order to correctly apply the change of variables in the double integral.

Step 2: Compute the Jacobian

The Jacobian matrix JJ of the transformation is the matrix of partial derivatives:

\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{pmatrix}$$ From the transformation equations, we compute the partial derivatives: $$\frac{\partial u}{\partial x} = -1, \quad \frac{\partial u}{\partial y} = 1$$ $$\frac{\partial v}{\partial x} = 1, \quad \frac{\partial v}{\partial y} = 1$$ Thus, the Jacobian matrix is: $$J = \begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}$$ The determinant of the Jacobian is: $$\text{det}(J) = (-1)(1) - (1)(1) = -2$$ The absolute value of the determinant is \(

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Multivariable Calculus
Change of Variables in Double Integrals

Formulas

Jacobian determinant for change of variables
Transformation formula for double integrals ∫∫_R f(x, y) dx dy = ∫∫_S f(g(u, v)) |det(J)| dudv

Theorems

Change of Variables in Double Integrals

Suitable Grade Level

Undergraduate Calculus

Related Recommendation

Evaluating Double Integral with Change of Variables Over a Trapezoidal Region

Evaluate Double Integral Using Change of Variables - Jacobian Method

Evaluating Double Integral with Transformation in Calculus

Double Integral Transformation: Calculate ∫∫(2x-y)cos(y-2x) dA over Triangle R

Evaluating Double Integral with Change of Variables for Parabolic Regions