Problem 54 - Applied Calculus

Question

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{2}^{3} \frac{x^{2}}{x^{3}-7} d x $$

Step-by-Step Solution

Answer

The evaluated integral is \( \frac{1}{3} \ln 20 \).

Step 1: Analyze the Integral

The definite integral to solve is \( \int_{2}^{3} \frac{x^{2}}{x^{3}-7} \, dx \). We will solve this integration through substitution.

Step 2: Substitution for Simplification

Let \( u = x^3 - 7 \). Then \( \frac{du}{dx} = 3x^2 \) or \( du = 3x^2 \, dx \). Hence, \( x^2 \, dx = \frac{1}{3} \, du \).

Step 3: Change of Limits

When \( x = 2 \), \( u = 2^3 - 7 = 1 \). When \( x = 3 \), \( u = 3^3 - 7 = 20 \). Therefore, the new limits of integration are from 1 to 20.

Step 4: Rewrite the Integral

Substitute \( u \) and the new differential into the integral: \( \int_{1}^{20} \frac{1}{3} \, \frac{1}{u} \, du \). This simplifies to \( \frac{1}{3} \int_{1}^{20} \frac{1}{u} \, du \).

Step 5: Integrate with Respect to u

The integral \( \int \frac{1}{u} \, du \) is \( \ln |u| + C \). Therefore, \( \frac{1}{3} \int_{1}^{20} \frac{1}{u} \, du = \frac{1}{3} [ \ln |u| ]_{1}^{20} \).

Step 6: Evaluate the Definite Integral

Evaluate \( \frac{1}{3} [ \ln |20| - \ln |1|] \). Since \( \ln |1| = 0 \), it simplifies to \( \frac{1}{3} \ln 20 \).

Step 7: Use a Graphing Calculator for Verification

Input the original integral \( \int_{2}^{3} \frac{x^{2}}{x^{3}-7} \ dx \) into a graphing calculator to confirm the result. It should also give \( \frac{1}{3} \ln 20 \) or approximately \(0.998797 \).

Key Concepts

Definite Integral

Definite integrals are fundamental tools in calculus used to find the area under a curve between two specified points along the x-axis. Unlike indefinite integrals, which yield a general function, definite integrals result in a numerical value. This is often used to calculate areas, volumes, and other quantities linked to the function being integrated.
To compute a definite integral:

• Identify the function to be integrated.
• Specify the limits of integration, in this case from 2 to 3.
• Carry out the integration process.
• Evaluate the antiderivative at the upper limit, subtract it from the value of the antiderivative evaluated at the lower limit.

The outcome, as in our exercise, turns into a number, representing the accumulated quantity of the function over the specified interval.

Substitution Method

The substitution method is a technique used to simplify complex integrals by introducing a new variable. It is similar to the reverse chain rule and is especially useful when you have a composite function within the integrand.
In our exercise:

• We set up the substitution by letting \( u = x^3 - 7 \). This changes the form of our integral by simplifying it.
• We find the derivative \( \frac{du}{dx} = 3x^2 \), which allows us to express \( dx \) in terms of \( du \).
• The original limits of integration, 2 and 3, are transformed into new limits for \( u \), changing from 1 to 20.
• The integral then becomes easier to manage, transforming to \( \int_{1}^{20} \frac{1}{3u} \, du \). This streamlined approach simplifies the calculation.

This method is powerful as it changes a complicated integral into a more straightforward problem, making the integration process easier to execute.

Graphing Calculator Verification

A graphing calculator can serve as an invaluable tool for verifying the results of an integral. After solving an integral manually, using a graphing calculator helps ensure the accuracy of your solution.
Here's how you can use it effectively:

• Input the original definite integral, here \( \int_{2}^{3} \frac{x^{2}}{x^{3}-7} \, dx \), directly into the calculator.
• Utilize the calculator's functions to compute the integral and compare this result to your manual calculations.
• This not only checks for errors but also boosts confidence in your computation skills.

In the given exercise, the graphing calculator confirmed the computed value of \( \frac{1}{3} \ln 20 \), approximating to \(0.998797\), reassuring the validity of the manual steps taken during integration.