Problem 64 - Biocalculus Calculus for the Life Sciences

Question

If \(f\) is continuous and \(\int_{0}^{9} f(x) d x=4,\) find \(\int_{0}^{3} x f\left(x^{2}\right) d x\)

Step-by-Step Solution

Answer

The integral evaluates to 2.

Step 1: Change of Variables

Let's start by making the substitution \( u = x^2 \). Consequently, \( du = 2x \, dx \) or \( dx = \frac{du}{2x} \). Our goal is to transform the original integral to be in terms of \( u \).

Step 2: Adjust Limits of Integration

The limits of the integral need to be converted as per the substitution \( u = x^2 \). When \( x = 0 \), \( u = 0^2 = 0 \). When \( x = 3 \), \( u = 3^2 = 9 \). Thus, the limits change from \( 0 \) to \( 9 \).

Step 3: Substitute and Simplify

Substitute \( u = x^2 \) and \( dx = \frac{du}{2x} \) into the integral:\[\int_{0}^{3} x f(x^2) \, dx = \int_{0}^{9} x f(u) \frac{du}{2x}.\]This simplifies to:\[\frac{1}{2} \int_{0}^{9} f(u) \, du.\]

Step 4: Evaluate the New Integral

Now use the given information \( \int_{0}^{9} f(x) \, dx = 4 \), which implies:\[\frac{1}{2} \int_{0}^{9} f(u) \, du = \frac{1}{2} \times 4 = 2.\]

Step 5: Conclusion

Therefore, the value of the integral \( \int_{0}^{3} x f(x^2) \, dx \) is 2, following our substitution and given information.

Key Concepts

continuous functions

In calculus, continuous functions play a crucial role, especially when dealing with integration. A function is considered continuous if, intuitively speaking, you can draw it without lifting your pencil off the paper. This means there are no holes, jumps, or breaks in the graph of the function.

Mathematically, a function \( f(x) \) is continuous at a point \( x = a \) if the following three conditions are satisfied:

• \( f(a) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) equals \( f(a) \).

Continuous functions have the Intermediate Value Theorem, making integration feasible over intervals. Thus, knowing that \( f \) is continuous ensures us that integral values like \( \int_{0}^{9} f(x) dx \) are calculable and predictable.

integral substitution

Integral substitution, also known as \(\textit{substitution rule}\), is one of the techniques employed to simplify integrals. This method involves changing the variable of integration to help transform a difficult integral into a simpler one.

The goal is to rewrite the integral in terms of a new variable, say \( u \), to simplify the integration process. Here is a simple breakdown of the steps involved in substitution:

• Select a substitution: For instance, in the exercise, \( u = x^2 \).
• Compute \( du \): Differentiate \( u \) to find \( du \), which in our case gives \( du = 2x \, dx \).
• Replace differential: Solve for \( dx \) in terms of \( du \), so here \( dx = \frac{du}{2x} \).
• Substitute in the integral: Every instance of \( x \) and \( dx \) gets replaced by expressions involving \( u \) and \( du \).

This substitution simplifies the process, helping you to evaluate the integral efficiently.

change of variables

The change of variables is a fundamental technique in calculus, used to simplify the evaluation of integrals by transforming variables. It is crucial when the direct integration process is complex or impractical.

In the provided example, instead of directly integrating \( \int_{0}^{3} x f(x^2) dx \), we perform a change of variables. By setting \( u = x^2 \), and finding the corresponding \( du \), the integration limits and the function transform accordingly. Here’s a concise guide on this process:

• Determine the new variable (\( u = x^2 \)) and express the original variable in its terms.
• Calculate \( du \) with respect to the original variable (\( du = 2x \, dx \)).
• Alter the integration limits: For \( x = 0 \), \( u = 0 \) and for \( x = 3 \), \( u = 9 \).
• Integrate using the new variable, making the computation simpler.

This technique allows transforming the integral bounds and functions for easier evaluation.

limits of integration

Limits of integration define the interval over which the integration occurs and are pivotal in defining definite integrals. When performing a variable substitution, adjusting these limits is crucial to ensure the integral accurately reflects the values over the intended interval.

Let's break down how limits are adjusted during substitution:

• Identify the original limits: In the exercise, they start as \(x = 0\) to \(x = 3\).
• Use the substitution to convert these into new limits: Here, substituting \( x = 0 \) gives \( u = 0^2 = 0 \) and \( x = 3 \) gives \( u = 3^2 = 9 \).
• Apply these new limits to the transformed integral, helping seamlessly transition from the original to the changed variables.

This careful adjustment ensures the definite integral calculates the area under the curve precisely across the desired range, using the new variable. By updating the limits of integration with the change of variables, one maintains the integrity and accuracy of the integration process.