Problem 9 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(7-20\). $$ \int_{1}^{e} \int_{1}^{e} \int_{1}^{e} \frac{1}{x y z} d x d y d z $$

Step-by-Step Solution

Answer

The value of the integral is 1.

Step 1: Setup the Integral

The given integral is a triple integral over the region where all three variables \(x\), \(y\), and \(z\) range from 1 to \(e\). The integrand is \(\frac{1}{xyz}\). The integral can be expressed as: \[\int_{1}^{e} \int_{1}^{e} \int_{1}^{e} \frac{1}{x y z} \, dx \, dy \, dz\]

Step 2: Evaluate the Innermost Integral

First, evaluate the integral with respect to \(x\). Since \(\frac{1}{x y z}\) is equivalent to \(\frac{1}{yz} \cdot \frac{1}{x}\), treat \(y\) and \(z\) as constants:\[\int_{1}^{e} \frac{1}{x} \, dx = \left[ \ln |x| \right]_{1}^{e} = \ln e - \ln 1 = 1\]

Step 3: Evaluate the Second Integral

Now, substitute the result of the innermost integral and evaluate the next integral with respect to \(y\): \[\int_{1}^{e} \frac{1}{y} \, dy = \left[ \ln |y| \right]_{1}^{e} = \ln e - \ln 1 = 1\]

Step 4: Evaluate the Outermost Integral

Substitute the result from the previous step into the outer integral with respect to \(z\):\[\int_{1}^{e} \frac{1}{z} \, dz = \left[ \ln |z| \right]_{1}^{e} = \ln e - \ln 1 = 1\]

Step 5: Compile the Results

Multiply the results gathered from each integral evaluation:\[1 \times 1 \times 1 = 1\]Thus, the value of the original triple integral is 1.

Key Concepts

Multiple Integrals

Multiple integrals extend the concept of a single integral to more than one variable or dimension. A triple integral is considered a type of multiple integral, used to integrate over a three-dimensional space.
In this context, the integral involves three variables: \(x\), \(y\), and \(z\). Each of these variables is integrated over the same interval from 1 to \(e\). Technically, this means that you are finding the volume under the surface defined by the integrand \(\frac{1}{xyz}\) within the bounded region.

When dealing with multiple integrals, especially triple integrals, the order of integration can significantly affect the complexity. However, unless the domain suggests otherwise, it generally doesn't affect the final result.

• Triple Integral: This specific type of multiple integral evaluates the integrand over a three-dimensional space.
• Order of Integration: Common order is \(\int dx \, dy \, dz\), but may vary depending on simplification needs.
• Domain Constraints: For each integral, the limits (here, 1 to \(e\)) remain constant and define the boundaries of their respective variables.

Understanding the concept of multiple integrals is essential for complex systems where one variable isn't enough to describe the system entirely.

Integral Calculus

Integral calculus is one of the main branches of calculus, focusing on the concept of integration. Its primary function is to find out the total accumulation of values, like area, volume, and other concepts that require summing infinitesimal data points.
In triple integrals, this concept extends to finding volumes under three-dimensional surfaces. The surface in question is defined by the function \(\frac{1}{xyz}\). Evaluating an integral, especially a triple integral, boils down to successive applications of integration rules for each dimension.

• Antiderivative: For each variable, find the antiderivative. For example, the antiderivative of \(\frac{1}{x}\) is \(\ln|x|\).
• Iterated Integration: Treat remaining variables as constants while integrating one variable at a time.
• Fundamental Theorem of Calculus: Apply this to compute definite integrals, which provide the net accumulation over an interval, key for solving the given problem.

Integral calculus is vital for solving real-world problems in physics and engineering by using such methods and principles.

Definite Integrals

Definite integrals calculate the net area under a curve within given limits, providing a concrete numerical result. In the exercise, the limits of integration are from 1 to \(e\) for each variable, yielding a set interval for evaluation.
A crucial aspect is setting these limits correctly. They define the region over which you are integrating and reflect the domain constraints relevant to the problem being solved. The exercise illustrates computing definite integrals successively for each variable, simplifying the problem into manageable parts that lead to the final result.

In context:

• Set Intervals: The bounds (1 to \(e\)) dictate the region for \(x\), \(y\), and \(z\).
• Sequential Evaluation: Start from the innermost integral and move outward.
• Final Result: Collate results from each step to obtain a singular outcome, here 1 as determined by multiplying the results from integrating \(x\), \(y\), and \(z\).

Understanding definite integrals is about recognizing them as tools for precise quantitative analysis of functions over specific intervals.