Problem 33 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int_{2}^{4} x^{\pi-1} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{4^{\pi} - 2^{\pi}}{\pi} \).

Step 1: Identify the Integral Function

We need to evaluate the definite integral \( \int_{2}^{4} x^{\pi-1} \, dx \). This is a simple power function \( x^{n} \), where \( n = \pi - 1 \).

Step 2: Use the Power Rule for Integration

The power rule for integration states that \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). Apply this to the integral: \( \int x^{\pi-1} \, dx = \frac{x^{\pi}}{\pi} + C \).

Step 3: Evaluate the Antiderivative at the Upper Limit

Substitute the upper limit 4 into the antiderivative: \( \frac{4^{\pi}}{\pi} \).

Step 4: Evaluate the Antiderivative at the Lower Limit

Substitute the lower limit 2 into the antiderivative: \( \frac{2^{\pi}}{\pi} \).

Step 5: Calculate the Definite Integral Result

Subtract the result at the lower limit from the result at the upper limit to find the definite integral: \( \frac{4^{\pi}}{\pi} - \frac{2^{\pi}}{\pi} \). This simplifies to \( \frac{4^{\pi} - 2^{\pi}}{\pi} \).

Step 6: Present the Final Answer

The value of the integral is \( \frac{4^{\pi} - 2^{\pi}}{\pi} \).

Key Concepts

Power Rule for Integration

One of the simplest and most important tools in calculus is the power rule for integration. This rule helps us to find an antiderivative for a power function. A power function is typically expressed as \( x^n \). The power rule states that if you want to find the integral of \( x^n \), you can use the formula:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), as long as \( n eq -1 \).

This means you add 1 to the exponent and then divide by the new exponent.
It's a simple yet powerful tool to integrate functions like \( x^{\pi-1} \), where \( n = \pi - 1 \).
Using the power rule, the integral turns into \( \frac{x^{\pi}}{\pi} + C \), without needing to memorize complex steps.

Antiderivative

The antiderivative is essentially the reverse of differentiation, giving us a function whose derivative is the original function. When we integrate a function like \( x^{\pi-1} \), we are finding the antiderivative.
For our integral \( \int x^{\pi-1} \, dx \), the antiderivative is \( \frac{x^{\pi}}{\pi} + C \).
Here, the constant 'C' is crucial since it represents any constant value that could have been present originally but vanished when differentiating.
In the context of definite integrals, this constant 'C' cancels out when we subtract the evaluated limits, simplifying our calculations.

Upper and Lower Limits

When calculating a definite integral, you work with two limits: the upper and lower limits, denoted here as 4 and 2 respectively. These limits define the interval over which you're integrating.
Think of these limits as boundaries. The process requires evaluating the antiderivative at these points to find specific values.
For the function \( \frac{x^{\pi}}{\pi} \), you do the following:

• Start by substituting the upper limit into the antiderivative: \( \frac{4^{\pi}}{\pi} \).
• Next, substitute the lower limit: \( \frac{2^{\pi}}{\pi} \).

Only by working with both limits can you find the precise numeric value that describes the area or other calculations related to the integral.

Calculating Area Under a Curve

A definite integral represents the signed area under a curve within specified limits. This area can be positive, negative, or zero, depending on the function's position relative to the x-axis.
For the integral \( \int_{2}^{4} x^{\pi-1} \, dx \), after finding the antiderivative and evaluating it at 4 and 2, you calculate the area by subtracting:

• \( \frac{4^{\pi}}{\pi} - \frac{2^{\pi}}{\pi} \)}.

With constants of integration gone, this subtraction reveals the definite integral's value, \( \frac{4^{\pi} - 2^{\pi}}{\pi} \).
The result quantifies the area under the curve of \( x^{\pi-1} \) from x = 2 to x = 4.
This geometric interpretation is a vital component of understanding calculus's power and utility.