Question

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int \frac{y}{x^{2}+y^{2}} d A\\\ &R: \text { triangle bounded by } y=x, y=2 x, x=2 \end{aligned} $$

Short answer

To find the double integral over the triangular region, the integral with respect to \(y\) first, then \(x\), is more convenient. Evaluating the integral: \[\int_{0}^{2} \int_{y/2}^{y} \frac{y}{x^{2}+y^{2}} dx dy\]is the final solution.

Step-by-step solution

Set Integral with respect to x First

Set up the integral with respect to x first, then y. The limits for \(x\) are from 0 to 2 according to \(x=2\) and those for \(y\) from \(x\) to \(2x\) according to \(y=x\) and \(y=2x\). This sets up the integral as:\[\int_{0}^{2} \int_{x}^{2x} \frac{y}{x^{2}+y^{2}} dy dx\]

Set Integral with respect to y First

Set up the integral with respect to \(y\) first, then \(x\). From the conditions \(y=x\) and \(y=2x\), \(x\) ranges from \(y/2\) to \(y\). Also, \(y\) ranges from 0 to 2 given the condition \(x=2\). This gives:\[\int_{0}^{2} \int_{y/2}^{y} \frac{y}{x^{2}+y^{2}} dx dy\]

Evaluate determined integrals

On comparison of both setups, integral with respect to \(y\) first is easier to evaluate since it avoids a complex integral computation. Hence, evaluate the integral: \[\int_{0}^{2} \int_{y/2}^{y} \frac{y}{x^{2}+y^{2}} dx dy\]to get the numerical value.

Key concepts

Double Integration

Double integration is a process used in calculus to calculate the volume under a surface for a given region in the plane, often denoted as a region \( R \). This operation involves integrating a function twice, typically in the context of two variables, x and y. In simpler terms, double integration lets us stack tiny volumes over a particular region to compute the total volume under a surface.

In our exercise, we need to find the integral of \(\frac{y}{x^2 + y^2}\) over a triangular region defined by three lines, \( y = x, \) \( y = 2x, \) and \( x = 2 \). We perform the integration process twice, once over each variable. Understanding the geometric interpretation of this region is crucial for setting up the integral correctly. Double integration helps us detect patterns and relationships between multiple variables as they change over the given area.

Order of Integration

The order of integration refers to the sequence in which the integrations are performed over multiple variables. For a double integral, this involves choosing whether to integrate with respect to x first or y first. Changing the order of integration is essential and can sometimes make an integral much easier to solve.

In the given problem, we first set up the integral by integrating with respect to x first, then y, updating our boundaries accordingly. However, we also evaluate it the other way around, integrating y first and then x.

• Integrating with x first involves limits from 0 to 2 for x and from x to 2x for y.
• Integrating with y first involves limits from y/2 to y for x and from 0 to 2 for y.

Choosing the more favorable order can simplify the problem significantly. In our case, integrating with respect to y first simplifies the evaluation because it prevents the integration of a more complex function in terms of x.

Integration Techniques

Integration techniques involve various methods that make solving integrals straightforward or even possible. These techniques include substitution, integration by parts, and switching the order of integration frames in multivariable calculus.

In solving the double integral in this exercise, switching the order of integration is an effective technique. It simplifies our task from evaluating a potentially complicated integral to a more manageable one by changing the variables. We also look to simplify integrals by narrowing the range of values that need to be considered in complex problems, making calculations less cumbersome.

In our specific example, solving the integral with changed limits reduces the complexity, making it easier to integrate the function \( \frac{y}{x^2 + y^2} \). These techniques are important tools in your math toolkit when dealing with integrals over regions like triangles or other non-rectangular domains.

Region of Integration

The region of integration is the area over which the double integration occurs. Determining this correctly is crucial as it influences both the order of integration and the limits that are set for the integral.

In the exercise, the region \( R \) is a triangle in the xy-plane bounded by the lines \( y = x, \) \( y = 2x, \) and \( x = 2 \) respectively. This triangular section forms the limits for our integrals:

• The line \( y = x \) sets boundaries for when \( y \) is at a minimum, while \( y = 2x \) establishes the maximum y-value for a given x.
• Finally, \( x = 2 \) is the vertical boundary which limits the entire region on the right-hand side.

Visualizing the region can aid in accurately setting these bounds and ensuring that we correctly capture the area under the function \( \frac{y}{x^2 + y^2} \). Understanding the region helps manage the assignment of proper limits, ensuring both the problem is solvable and the results obtained are meaningful.