Question

State why it is difficult/impossible to integrate the iterated integral in the given order of integration. Change the order of integration and evaluate the new iterated integral. $$ \int_{0}^{1} \int_{y}^{1} \frac{2 y}{x^{2}+y^{2}} d x d y $$

Short answer

Changing the order of integration and solving yields \( \ln(2) \).

Step-by-step solution

Evaluate Difficulty with Original Order

The given integral is \( \int_{0}^{1} \int_{y}^{1} \frac{2 y}{x^{2}+y^{2}} \, dx \, dy \). In the current order, integrating with respect to \( x \) first is complex because the expression \( x^2 + y^2 \) in the denominator suggests an arctangent component, which cannot be easily integrated over the bounds \( x = y \) to \( x = 1 \). Moreover, substituting \( x = y \) creates a singularity, making integration problematic.

Visualize Region of Integration

To change the order of integration, determine the region of integration. The current bounds are \( x = y \) to \( x = 1 \) and \( y \) from 0 to 1. Therefore, the region is a triangular area bounded by \( y = 0 \), \( x = 1 \), and \( x = y \).

Change Order of Integration

To change the order of integration, switch the bounds to integrate with respect to \( y \) first and then \( x \). For each fixed \( x \), \( y \) ranges from 0 to \( x \). Thus, the new integral becomes \( \int_{0}^{1} \int_{0}^{x} \frac{2 y}{x^{2}+y^{2}} \, dy \, dx \).

Integrate with Respect to y

First, consider the inner integral \( \int_{0}^{x} \frac{2 y}{x^{2} + y^{2}} \, dy \). Use the substitution \( u = y^2 \), then \( du = 2y \, dy \). Therefore, the integral becomes \( \int_{0}^{x^2} \frac{1}{x^2 + u} \, du \). This simplifies to \( \ln(x^2 + u) \) evaluated from 0 to \( x^2 \), resulting in \( \ln(x^2 + x^2) - \ln(x^2) = \ln(2) \).

Integrate with Respect to x

Now integrate the result of the previous step with respect to \( x \): \( \int_{0}^{1} \ln(2) \, dx \). Since \( \ln(2) \) is constant with respect to \( x \), the integral is simply \( \ln(2) \cdot x \), evaluated from 0 to 1. This results in \( \ln(2) \cdot 1 - \ln(2) \cdot 0 = \ln(2) \).

Conclusion: Evaluate the New Iterated Integral

The new order of integration transformed the problem to an easily solvable one, resulting in a final answer of \( \ln(2) \).

Key concepts

Order of Integration

The order of integration refers to the sequence in which multiple integrals are evaluated. For iterated integrals, the choice of performing integration with respect to one variable before another can significantly impact the complexity of the computation. In some cases, as seen in the original problem, integrating with respect to the variable \( x \) first led to a complex expression involving an arctangent function. This kind of situation often suggests that re-evaluating the sequence of integration might simplify your work.

By changing the order of integration, we can often transform a difficult integral into an easier one. In this specific integral, once the order was swapped, integrating with respect to the new sequence became straightforward. Understanding the effect of the order of integration is crucial, as it enhances your ability to handle complex integrals efficiently.

Singularity in Integrals

Singularities occur in integrals when the function becomes undefined or discontinuous at some point within the limits of integration. In this exercise, the original order of integration produced a singularity when \( x = y \).

This happens because, at that boundary, the denominator \( x^2 + y^2 \) could approach zero, complicating the process. Singularities make integration directly through standard methods almost impossible, as they require careful handling or strategic manipulation of the region or order of integration.

Recognizing potential singularities is important because it alerts you to reevaluate your method, possibly by modifying integration order or employing approximation techniques to avoid computational errors.

Substitution Method

The substitution method is a powerful tool in calculus, especially for transforming difficult integrals into simpler forms. In this exercise, substitution played a key role when changing the variable during integration with respect to \( y \).

The substitution \( u = y^2 \) helped simplify the expression inside the integral to \( \int \frac{1}{x^2 + u} \, du \). This transformation is key in dealing with complex integrals, where direct integration is challenging. The beauty of substitution lies in its ability to align difficult functions to familiar forms like logarithmic or trigonometric integrals, facilitating simpler evaluation.

Mastering substitution can open pathways to solving many integrals that seem unsolvable at first glance.

Region of Integration

The region of integration defines the area over which integration occurs, influencing both the complexity and feasibility of the process. Initially, in this problem, the region was constrained by conditions \( x = y \) to \( x = 1 \) and \( y = 0 \) to \( y = 1 \), forming a triangular shape.

Visualizing and understanding this region are crucial steps when deciding to change the order of integration. By visualizing the region, you can adjust the boundaries to fit the new variable order, simplifying the integral's evaluation.

Good comprehension of the region of integration helps avoid mistakes and can illuminate opportunities to transform and solve otherwise challenging integrals. This flexibility is essential in advanced calculus tasks where varying regions and boundaries are commonplace.