Problem 14 - Advanced Engineering Mathematics

Question

Evaluate the double integral over the region \(R\) that is bounded by the graphs of the given equations. Choose the most convenient order of integration. $$ \iint_{R}(x+1) d A ; y=x, x+y=4, x=0 $$

Step-by-Step Solution

Answer

The value of the double integral is \( \frac{22}{3} \).

Step 1: Understand the Region

The region \( R \) is bounded by the lines \( y = x \), \( x + y = 4 \), and \( x = 0 \). To find the region, sketch these lines over a coordinate system. The line \( y = x \) intersects \( x + y = 4 \) at \( (2, 2) \). The line \( x = 0 \) is the y-axis. Thus, the region is a triangular area with vertices at \( (0,0) \), \( (0,4) \), and \( (2,2) \).

Step 2: Decide the Order of Integration

Considering the boundaries \( x = 0 \) to \( x = 2 \) and \( y = x \) to \( y = 4 - x \) (these are the y-limits for a given x), integrating with respect to \( y \) first followed by \( x \) is convenient for this region.

Step 3: Set Up the Double Integral

Set up the integral as \( \int_{x=0}^{2} \int_{y=x}^{4-x} (x+1) \, dy \, dx \). This setup accounts for the vertical strip method, projecting between the y-boundaries within the given x-boundaries.

Step 4: Integrate with Respect to y

Evaluate the inner integral \( \int_{y=x}^{4-x} (x+1) \, dy \). This results in \( (x+1)y \Bigg|_{y=x}^{4-x} \), which simplifies to \( (x+1)(4-x) - (x+1)x = (3-x)(x+1) \).

Step 5: Simplify the Resulting Expression

Expand and simplify \( (3-x)(x+1) \) to \( (3-x)((x+1)) = 3x + 3 - x^2 - x = 2x + 3 - x^2 \).

Step 6: Integrate with Respect to x

Now evaluate the outer integral \( \int_{0}^{2} (3 - x^2 + 2x) \, dx \). Integrate term by term: \( \int (3 - x^2 + 2x) \, dx = 3x - \frac{x^3}{3} + x^2 \Bigg|_0^2 \).

Step 7: Evaluate the Final Expression

Substitute \( x = 2 \) into the expression: \( 3(2) - \frac{(2)^3}{3} + (2)^2 = 6 - \frac{8}{3} + 4 \). Simplifying, \( 10 - \frac{8}{3} = \frac{30}{3} - \frac{8}{3} = \frac{22}{3} \).

Step 8: State the Answer

The value of the double integral is \( \frac{22}{3} \).

Key Concepts

Order of Integration

Choosing the most convenient order of integration is crucial when evaluating a double integral. This means deciding whether to integrate with respect to one variable first, and then the other. Typically, the choice depends on the geometry of the region and how the limits of integration simplify the problem.
In our example, the problem was solved by integrating with regard to \( y \) first, from \( y = x \) to \( y = 4 - x \), followed by \( x \), ranging from 0 to 2. This choice suits the bounded region since it allows us to handle simpler limits that vary linearly.
To decide the order, one should consider:

• Which variable simplifies the limits or the integrand?
• Does the order lead to simpler calculations?

Making the right choice helps in minimizing computational complexity and avoiding unnecessary mistakes.

Bounded Region

In a double integral, understanding the bounded region \( R \) is fundamental. In this context, the region is a part of the coordinate plane enclosed by given curves or lines.Identifying this region properly ensures the correct limits for the integral.
For the given problem, the bounded region is formed by the lines \( y = x \), \( x + y = 4 \), and \( x = 0 \). Sketching these lines reveals a triangular area with vertices at \( (0,0) \), \( (0,4) \), and \( (2,2) \).
To accurately sketch:

• Plot each line on the coordinate plane.
• Determine the points of intersection or where these lines meet.
• Identify closed shapes formed by intersections as the bounded region.

Properly identifying this region is crucial for setting the correct integration limits.

Integral Evaluation

The process of evaluating a double integral involves calculating the exact value of the integral across a specified region. Once the region and order of integration are clear, the integration takes place over two steps, corresponding to each variable.
The original problem first integrated over \( y \), and the inner integral was found to be \( (3-x)(x+1) \). This was achieved by substituting the boundaries into the indefinite integral formed by \( (x+1)y \).
After simplifying and evaluating this inner integral, the next step is to carry out the outer integration with respect to \( x \), using these results to find the total integral value.
Steps to ensure successful evaluation:

• Set up the integral with the correct boundaries.
• Perform the inner integration and simplify where possible.
• Evaluate the outer integral using the outcome of the first.
• Substitute any remaining variable limits and simplify to reach the final result.

This methodical approach will reveal the precise integral for the bounded region.

Coordinate Geometry

Coordinate geometry provides the framework needed to understand the properties of curves and shapes in a plane, using their equations. It enables the examination of how these curves intersect and form bounded regions, essential for evaluating double integrals.
In the provided example, the lines \( y = x \), \( x + y = 4 \), and \( x = 0 \) were sketched in a coordinate system to identify their intersections. The line \( y = x \) intersects \( x + y = 4 \) at \( (2, 2) \), providing key information for the problem's region.
Key tasks for handling coordinate geometry within integrals include:

• Sketching equations on the coordinate plane for visual guidance.
• Determining intersection points to outline regions.
• Understanding the spatial relationships of multiple lines or curves.

By mastering these concepts, one can effectively solve problems involving double integrals over complex regions defined via coordinate geometry.