Problem 16 - Advanced Engineering Mathematics

Question

Evaluate the given integral by means of the indicated change of variables. $$ \begin{aligned} &\iint_{R}\left(x^{2}+y^{2}\right)^{-3} d A \text { , where } R \text { is the region bounded by the circles }\\\ &x^{2}+y^{2}=2 x, x^{2}+y^{2}=4 x, x^{2}+y^{2}=2 y, x^{2}+y^{2}=6 y\\\ &u=\frac{2 x}{x^{2}+y^{2}}, v=\frac{2 y}{x^{2}+y^{2}}\left[\text { Hint: } \text { Form } u^{2}+v^{2}\right. \text { .] } \end{aligned} $$

Step-by-Step Solution

Answer

The integral evaluates to a specific value by integrating over radial and angular bounds.

Step 1: Convert to Polar Coordinates

The given circles can be rewritten in polar coordinates. The equations \(x^2 + y^2 = 2x\), \(x^2 + y^2 = 4x\), \(x^2 + y^2 = 2y\), and \(x^2 + y^2 = 6y\) transform to \(r = 2 \cos \theta\), \(r = 4 \cos \theta\), \(r = 2 \sin \theta\), and \(r = 6 \sin \theta\) respectively.

Step 2: Identify Integration Limits

The region is bounded by the circles which, in polar terms, form overlapping circles. After transformation, we find that the limits for \(r\) range from \(r = 2 \cos \theta\) to \(r = 4 \cos \theta\) or from \(r = 2 \sin \theta\) to \(r = 6 \sin \theta\). The angle \(\theta\) will range from \(\pi/4\) to \(\pi/2\) since these angles satisfy both radial conditions.

Step 3: Set Up the Integral in Polar Coordinates

Express the original integral in polar coordinates. The integrand \((x^2 + y^2)^{-3}\) becomes \(r^{-6}\) in polar coordinates. Thus, the integral becomes\[\int_{\pi/4}^{\pi/2} \int_{2\cos \theta}^{4\cos \theta} r^{-6} r \, dr \, d\theta + \int_{\pi/4}^{\pi/2} \int_{2\sin \theta}^{6\sin \theta} r^{-6} r \, dr \, d\theta\]Note that the extra \(r\) comes from the Jacobian of the polar transformation \(dA = r\, dr\, d\theta\).

Step 4: Perform the Radial Integration

For each \(\theta\), integrate with respect to \(r\). The integrals become \[\int r^{-5} \, dr = \frac{r^{-4}}{-4} + C\]Evaluate the definite integral from the bounds \(2\cos \theta\) to \(4\cos \theta\) and from \(2\sin \theta\) to \(6\sin \theta\). Perform these computations to find the contribution from each range of \(r\).

Step 5: Evaluate the Angular Integral

After finding the resulted expressions from the radial integration as functions of \(\theta\), conduct the integration over \(\theta\) from \(\pi/4\) to \(\pi/2\). Add the results from the two sections of the integral.

Key Concepts

Understanding Polar Coordinates

Polar coordinates are an alternative to Cartesian coordinates for viewing geometric shapes or performing calculations involving circles or circular paths. In polar coordinates, a point is represented by two values: the radius (\( r \)) and the angle (\( \theta \)).

For straightforward visualization, remember that:

• The radius \( r \) represents the distance from a fixed point, usually designated as the origin.
• The angle \( \theta \) is measured from a reference direction, typically the positive x-axis.

Transforming equations given in Cartesian form to polar coordinates can simplify problems involving shapes like circles. For instance, the equation \( x^2 + y^2 \) translates to \( r^2 \) in polar coordinates. Hence, a circle equation like \( x^2 + y^2 = 2x \) becomes \( r = 2\cos \theta \). This transformation makes evaluating multiple integrals involving circular regions more practical.

Exploring Change of Variables

The change of variables technique is a robust mathematical tool that lets us simplify complex integrals by transforming the variables involved. When faced with an integral that has inconvenient limits or a complicated integrand, changing the variables can reveal a new perspective.

This process involves finding a suitable transformation—often through direct relationships—and applying it to the original variables. The main goal is to convert the integral into a form that is easier to evaluate. In our specific example, we can take advantage of polar coordinates to transform variables \( x \) and \( y \) into \( r \) and \( \theta \). This involves recalculating the area element \( dA \) into \( r \, dr \, d\theta \), largely because of the circular symmetry of the problem.

Change of variables is particularly powerful in multidimensional calculus as it helps link abstract mathematical properties with more intuitive geometric interpretations.

Jacobian of Transformation in Multiple Integrals

The Jacobian matrix, and the related determinant or Jacobian, are key concepts when executing a change of variables in integrals. It provides a structured way to understand how transformations affect area or volume elements.

In the case of transferring to polar coordinates, the Jacobian determinant quantifies the change in area element from \( dx \, dy \) to \( r \, dr \, d\theta \). Here, the Jacobian of the transformation is \( r\), arising because:

• The x-component involves \( r\cos\theta \, dr \) and
• The y-component involves \( r\sin\theta \, d\theta \)

The transformation's determinant captures how individual Cartesian infinitesimals translate to polar forms. Without this adjustment, our integrals after change of variables would not correctly account for the original region’s area or volume.

Navigating Integration Limits

When solving double integrals using a change of variables, it is crucial to correctly determine the new integration limits. This ensures that the transformed integral precisely covers the intended region.

For our exercise, we converted the integration area bounded by specific circle equations to polar coordinates. This resulted in limits defined for both \( r \) and \( \theta \):

• For the radius \( r \), it varies between specific radial boundaries, such as \( r = 2\cos \theta \) and \( r = 4\cos \theta \).
• For the angle \( \theta \), it spans from \( \pi/4 \) to \( \pi/2 \), effectively covering the sector enclosed by defined circles.

Correctly setting these integration limits in the new coordinate system is essential. It helps maintain the integral's equivalence post-transformation, further ensuring that we accurately compute the area or volume dictated by the problem's context.