Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x y\\\ &R: \text { rectangle with vertices }(0,0),(4,0),(4,2),(0,2) \end{aligned} $$

Step-by-Step Solution

Answer

The average value of \(f(x, y) = xy\) over the region \(R\) is 1.

Step 1: Compute the area

Before we can compute the average value, we need to know the total area of the region R. Given the vertices of the rectangle, we can compute it's area A as the product of its length and width. This gives us \(A = (4-0)*(2-0) = 8\).

Step 2: Set up the double integral

The average value formula for a function \(f(x, y)\) over a region \(R\) is given by \[\frac{1}{A} \int\int_{R} f(x, y) dA\] After inserting the function \(f(x,y)=xy\) and the area \(A=8\), as well as the integration limits (0 to 4 for x and 0 to 2 for y), we get \[\frac{1}{8} \int_0^4 \int_0^2 xy \,dy\, dx\].

Step 3: Compute the inner integral

Next, we compute the inner integral, for which y is the variable of integration: \[\frac{1}{8} \int_0^4 [\frac{1}{2}x*y^2]_0^2 dx = \frac{1}{8} \int_0^4 x*4 \,dx = \frac{1}{8} \int_0^4 4x \,dx = \int_0^4 x \,dx\].

Step 4: Compute the outer integral

Now, a single variable integration is remaining: \[\int_0^4 x \,dx = [\frac{1}{2}*x^2]_0^4 = 8\].

Step 5: Evalutate the integral and find average

Finally, we get the average value of \(f\) on \(R\) by evaluating the integral: \(Average = \frac{1}{A} * \int_0^4 x \,dx = \frac{1}{8} * 8 = 1\).

Key Concepts

Average Value of a Function

To find the average value of a function over a given region, we want to determine what the "mean" of the function's output is throughout that entire area. This is done by integrating the function across the region and dividing by the region's total area. Given a function, like \( f(x, y) = xy \), and a specific region \( R \), we use the formula:

• Average Value = \( \frac{1}{A} \int \int_{R} f(x, y) \, dA \)

where \( A \) is the area of the region \( R \). When integrating over the region, it's crucial to utilize the correct bounds and perform the integration properly for each variable involved. Once the integration over \( R \) is done, you divide the resulting value by \( A \) to obtain the function's average output over this region. Understanding this process is key in multivariable calculus as it forms a basis for many applications, including physics and engineering.

Region of Integration

A region of integration is essentially the area over which a double integral is evaluated. In our example, the region is a rectangle determined by its vertices, given as points

• \((0,0),(4,0),(4,2),(0,2)\).

These points mark the corners of the rectangle and help define the limits of integration in the Cartesian plane. To identify a region of integration, you should understand the bounds of each variable involved:

• \( x \) ranges from 0 to 4
• \( y \) ranges from 0 to 2

It's important to note that the boundaries for \( x \) and \( y \) can be more complex for different regions, such as circular or triangular regions. But in the case of a rectangle, it's straightforward to determine these linear limits. Getting comfortable with analyzing different regions provides a foundation for more complex calculus challenges.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, provide a straightforward way to describe locations in space using two numbers. Each point is determined by an \( x \)-coordinate and a \( y \)-coordinate, detailing a unique position on a plane. This system is especially useful for integration as it aligns well with the natural perpendicular axes, \( x \) and \( y \), making it easy to define regions such as rectangles. In our exercise, the rectangle's vertices help define the space we are interested in, tying directly to the \( x \)- and \( y \)-coordinates. These coordinates allow for effective breakdown and computation of double integrations, especially when dealing with rectangles. Using rectangular coordinates simplifies the integration process, particularly when the limits of integration are constant, helping you deal directly with bounding straight-line regions.

Multivariable Calculus

Multivariable calculus is an extension of calculus involving functions of more than one variable. Whereas traditional calculus focuses on functions of a single variable, multivariable calculus introduces partial derivatives, multiple integrals, and – as in our exercise – specifies calculations over specified regions.Central concepts include understanding how to:

• Handle functions of several variables.
• Set up integrals over complex regions.
• Utilize derivatives to evaluate rates of change in multiple directions.

In the context of our problem, multivariable calculus allows us to compute the average value of a function \( f(x, y) \) over a defined rectangular region. By breaking this down into iterative integrations, we take slices through one variable while holding the other constant, progressing from inner to outer integration. Mastering these approaches provides powerful tools for applications in various scientific fields, including engineering and physics, where analyzing relationships and changes in multiple dimensions is essential.