Problem 49 - Calculus Single Variable

Question

In each of Exercises \(43-52\) calculate the average of the given expression over the given interval. $$ 4 x \ln (x) \quad 1 \leq x \leq e $$

Step-by-Step Solution

Answer

The average value is \( \frac{e^2 + 1}{e - 1} \).

Step 1: Set up the average value formula

To find the average value of a function on an interval, use the formula: \[\text{Average} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx\] where \( f(x) = 4x \ln(x) \), \( a = 1 \), and \( b = e \).

Step 2: Write the integral expression

Insert the function and limits of integration into the average value formula: \[\text{Average} = \frac{1}{e-1} \int_{1}^{e} 4x \ln(x) \, dx\]

Step 3: Compute the integral

The integral \( \int 4x \ln(x) \, dx \) requires integration by parts. Let \( u = \ln(x) \) and \( dv = 4x \, dx \). Then \( du = \frac{1}{x} \, dx \) and \( v = 2x^2 \). Apply integration by parts: \[ \int 4x \ln(x) \, dx = [2x^2 \ln(x)] - \int 2x^2 \cdot \frac{1}{x} \, dx \]\[ = 2x^2 \ln(x) - 2\int x \, dx \]\[ = 2x^2 \ln(x) - x^2 \]Evaluate this from 1 to \( e \).

Step 4: Evaluate the integral expression

Substitute the limits into the expression: \[\left[2x^2 \ln(x) - x^2\right]_{1}^{e} = \left(2e^2 \ln(e) - e^2\right) - \left(2\cdot1^2 \ln(1) - 1^2\right)\]Knowing that \( \ln(e) = 1 \) and \( \ln(1) = 0 \), simplify: \[2e^2 \cdot 1 - e^2 = e^2 \]Hence, \[[e^2] - [0 - 1] = e^2 - (-1) = e^2 + 1 \]

Step 5: Calculate the average value

Plug the result of the integral back into the average value formula:\[\text{Average} = \frac{1}{e-1}(e^2 + 1) = \frac{e^2 + 1}{e - 1}\] Simplify if needed, but this is the average value.

Key Concepts

Average Value of a Function

The average value of a function gives us a sense of the function's overall behavior over a certain interval. It is a way to summarize the function's behavior between two points, just like computing the average of numbers gives us an idea of their central tendency. To compute this, we use the formula:\[\text{Average} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx\]where:

• \(a\) and \(b\) are the lower and upper limits of the interval,
• \(f(x)\) is the function in question,
• The integral \(\int_{a}^{b} f(x) \, dx\) is the area under the curve of the function from \(a\) to \(b\).

In our exercise, we want to find the average value of \(4x \ln(x)\) from \(x = 1\) to \(x = e\). By plugging these values into the formula, we can evaluate the integral and compute the average using the resulting value.

Definite Integral

The definite integral is a fundamental concept in calculus that represents the signed area under a curve between two points. It combines the function's values over an interval to give us a single number, which can represent things like total area, displacement, or in our case, part of the calculation for the average value. A definite integral is written as:\[\int_{a}^{b} f(x) \, dx\]where:

• \(a\) and \(b\) are boundaries of the interval,
• \(f(x)\) is the function to be integrated,
• The integral gives the net area between the curve \(f(x)\) and the x-axis from \(x = a\) to \(x = b\).

In the exercise, we used the definite integral \(\int_{1}^{e} 4x \ln(x) \, dx\), which required solving by the technique of integration by parts, revealing the function's integral value crucial to determining the average value.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithmic function with the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. It is widely used in mathematics due to its natural properties in calculus and exponential growth models.Key characteristics of the natural logarithm:

• The natural logarithm of 1 is 0: \( \ln(1) = 0 \).
• \( \ln(e) \, is \, equal \, to \, 1 \) since \(e^1 = e\).

In the context of the exercise, the term \( \ln(x) \) appears in the function \(4x \ln(x)\), and we need to understand the properties of \(\ln(x)\) to correctly apply integration techniques such as integration by parts. The understanding of natural logarithms helps simplify expressions and evaluate limits in calculus.