Problem 18 - Advanced Engineering Mathematics

Question

Evaluate the given integral by means of the indicated change of variables. \(\begin{aligned} &\iint_{R}\left(x^{2}+y^{2}\right) \sin x y d A, \text { where } R \text { is the region bounded by the }\\\ &\text { graphs of } x^{2}-y^{2}=1, x^{2}-y^{2}=9, x y=2, x y=-2\\\ &u=x^{2}-y^{2}, v=x y \end{aligned}\)

Step-by-Step Solution

Answer

Transform using the variables and integrate over 1 to 9 for \( u \) and -2 to 2 for \( v \).

Step 1: Define the change of variables

Let \( u = x^2 - y^2 \) and \( v = xy \). We will change variables from \( (x, y) \) to \( (u, v) \). This requires finding \( x \) and \( y \) in terms of \( u \) and \( v \).

Step 2: Express x and y in terms of u and v

Solve the system of equations: \( u = x^2 - y^2 \) and \( v = xy \). These can be expressed as \( x = r \cosh(\theta) \) and \( y = r \sinh(\theta) \) for appropriate \( r \). Use the identity \( x^2 - y^2 = u \) and \( xy = v \) to find expressions for \( x \) and \( y \) in terms of \( u \) and \( v \).

Step 3: Compute the Jacobian of the transformation

Find the Jacobian \( J \) of the transformation from \( (x, y) \) to \( (u, v) \). Calculate the partial derivatives \( \frac{\partial(x, y)}{\partial(u, v)} \). The Jacobian, \( J = \left| \frac{\partial(x, y)}{\partial(u, v)} \right| \), should be calculated as \( \det \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} \).

Step 4: Determine the transformed region

The original bounds given are \( x^2 - y^2 = 1 \) to \( x^2 - y^2 = 9 \) and \( xy = 2 \) to \( xy = -2 \). In \( (u, v) \) coordinates, the region is \( 1 \leq u \leq 9 \) and \( -2 \leq v \leq 2 \).

Step 5: Rewrite the integral in terms of u and v

Rewrite the integrand \( (x^2 + y^2) \sin(xy) \) in terms of \( u \) and \( v \). The term \( x^2 + y^2 = \sqrt{u^2 + 4v^2} \) and \( \sin(xy) = \sin(v) \). The integral becomes \( \iint_{1 \leq u \leq 9, -2 \leq v \leq 2} \sqrt{u^2 + 4v^2} \sin(v) \cdot J \cdot dudv \), where \( J \) is the Jacobian.

Step 6: Evaluate the integral

Substitute the bounds and the integrand expressed in terms of \( u \) and \( v \) into the double integral and evaluate. This step involves performing the integration over \( v \) first from \( -2 \) to \( 2 \), and then over \( u \) from \( 1 \) to \( 9 \). Simplify the expression and solve the integral, noting that if the integrand is complex, numerical methods might be used.

Key Concepts

Jacobian Matrix

The Jacobian Matrix is crucial when performing a change of variables in double integrals. It links the differentials of the original coordinates to the new transformed ones, ensuring that the integral's value remains consistent.
A Jacobian Matrix is constituted by partial derivatives and is of the form:

• For a transformation from \((x, y)\) to \((u, v)\), the Jacobian Matrix is \( J = \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix}\)
• The determinant of this matrix, \( \left| J \right| \), quantifies the area adjustment needed when transitioning between coordinate systems.

To compute \( J \), solve for expressions of \(x\) and \(y\) in terms of \(u\) and \(v\), then derive the necessary partial derivatives. This understanding aids in accurately setting up and evaluating the integral in new coordinates.

Transformation of Coordinates

Transformation of coordinates is a method utilized to simplify complex integrals by changing the variables in a coherent manner. Here's how it works:

• The process begins with determining new variables, say \(u\) and \(v\), that simplify the region of integration or the integrand itself.
• For example, in the exercise, the transformation \(u = x^2 - y^2\) and \(v = xy\) was used. This specific choice might align with the geometrical properties of the region \(R\), simplifying the integral bounds and the integrand.

Once variables are transformed, their relationships redefine the region of integration. This approach often turns difficult integrations into simpler or more recognizable problems, making evaluations more feasible.

Multiple Integrals

Multiple Integrals extend the idea of integration to functions of multiple variables. They allow computation over a defined region, often in two or three dimensions.
Here's a breakdown of how they work:

• Double integrals, such as \(\iint_R f(x, y) \,dA\), involve integrating a function \(f(x, y)\) over a region \(R\).
• The result is affected by both changes within \(R\) and by transformations of \(f(x, y)\) alone.

Performing multiple integrals often involves a step-wise approach: handle one variable at a time, mindful of corresponding limits or bounds. This methodology is essential for applications in physics and engineering, where calculations over various domains are frequent. Changing variables with methods like using the Jacobian makes these calculations manageable.