Problem 37 - Thomas Calculus

Question

In Exercises \(37-40,\) find the average value of \(F(x, y, z)\) over the given region. $$ \begin{array}{l}{F(x, y, z)=x^{2}+9 \text { over the cube in the first octant bounded by }} \\ {\text { the coordinate planes and the planes } x=2, y=2, \text { and } z=2}\end{array} $$

Step-by-Step Solution

Answer

The average value is \(\frac{31}{3}\).

Step 1: Understand the Region

The region where we need to find the average value is a cube in the first octant bounded by the coordinate planes and the planes \(x = 2\), \(y = 2\), and \(z = 2\). This means our limits for integration are 0 to 2 for \(x\), \(y\), and \(z\).

Step 2: Set Up the Volume Integral

To find the average value, we need the integral of \(F(x, y, z) = x^2 + 9\) over the cube. Set up the triple integral as:\[\int_0^2 \int_0^2 \int_0^2 (x^2 + 9) \, dx \, dy \, dz\]

Step 3: Compute the Inner Integral with respect to x

Focus on the inner integral first:\[\int_0^2 (x^2 + 9) \, dx = \left[ \frac{x^3}{3} + 9x \right]_0^2 \]Calculate this to get:\[\left( \frac{8}{3} + 18 \right) - (0 + 0) = \frac{8}{3} + 18 = \frac{62}{3}\]

Step 4: Compute the Second Integral with respect to y

Substitute the evaluated integral from Step 3 into the next layer:\[\int_0^2 \frac{62}{3} \, dy = \left[ \frac{62}{3}y \right]_0^2 = \frac{62}{3} \times 2 = \frac{124}{3}\]

Step 5: Compute the Outer Integral with respect to z

Now, evaluate the last integral:\[\int_0^2 \frac{124}{3} \, dz = \left[ \frac{124}{3}z \right]_0^2 = \frac{124}{3} \times 2 = \frac{248}{3}\]

Step 6: Calculate the Volume of the Cube

The volume of the cube is given by the product of its sides, which is \[2 \times 2 \times 2 = 8\].

Step 7: Determine the Average Value

Finally, calculate the average value of \(F(x, y, z)\) by dividing the result of the integral by the volume:\[\text{Average} = \frac{248/3}{8} = \frac{248}{24} = \frac{31}{3}\]

Key Concepts

Triple Integrals

Triple integrals are a powerful tool in calculus for evaluating functions over three-dimensional regions. They extend the concept of double integrals, which deal with two-dimensional areas, to three dimensions.
Triple integrals are particularly valuable in calculating volumes, mass, and other properties of 3D objects.

• The notation for a triple integral is \(\int \int \int f(x, y, z) \ dx \, dy \, dz\), where the function \(f(x, y, z)\) is evaluated over a three-dimensional space.
• Triple integrals typically involve determining the limits for each variable \(x, y,\) and \(z\) according to the region of integration.

Understanding how to set the limits correctly is crucial. In our exercise, the limits for \(x\), \(y\), and \(z\) are from 0 to 2 due to the constraints of the cube in the first octant. From there, each layer of integration is handled successively, always working from the innermost integral to the outermost.

Average Value of a Function

The average value of a function provides a way to find a single representative value from a distribution of function values over a specific region. It can give insights into the function's behavior on the entire domain.

• To calculate the average value of a function \(f(x, y, z)\) over a volume, you first compute the triple integral over the region, which gives you the total sum of the function's values throughout that volume.
• Then, divide this sum by the volume of the region to find the average value.

In the case of the exercise provided, the function \(F(x, y, z) = x^2 + 9\) was integrated over a specific cube, and its total was divided by the cube's volume \(8\). The result was \( \frac{31}{3} \), which is the average value of the function over that particular cubic region.

Multivariable Calculus

Multivariable calculus is an extension of calculus that deals with functions of several variables. Unlike single-variable calculus, which operates on functions of one variable, multivariable calculus lets us work with functions that depend on two or more variables. It is essential in fields where phenomena depend on multiple factors, such as physics and engineering.

• Key operations in multivariable calculus include partial derivatives, multiple integrals (like triple integrals), and extremely useful concepts such as gradients and divergence.
• Multivariable calculus allows us to analyze how a function changes along different paths in space and is crucial to understanding complex systems.

The triple integral in the provided exercise is a classic example of multivariable calculus. We are integrating a function that depends on three variables \(x, y,\) and \(z\) across a defined spatial domain, which involves understanding and manipulating concepts well beyond the scope of single-variable calculus.