Problem 65 - Biocalculus Calculus for the Life Sciences

Question

If \(f\) is continuous and \(\int_{0}^{4} f(x) d x=10,\) find \(\int_{0}^{2} f(2 x) d x\)

Step-by-Step Solution

Answer

The value of the integral is 5.

Step 1: Substitute the Variable

Let us make a substitution of the variable to solve the integral \( \int_{0}^{2} f(2x) \, dx \). Define \( u = 2x \) such that \( du = 2 \, dx \). Therefore, \( dx = \frac{1}{2} \, du \). When \( x = 0 \), \( u = 0 \), and when \( x = 2 \), \( u = 4 \). This changes the bounds of integration.

Step 2: Transform the Integral

Substitute \( dx = \frac{1}{2} \, du \) and adjust the limits of integration: \[ \int_{0}^{2} f(2x) \, dx = \int_{0}^{4} f(u) \left(\frac{1}{2}\right) \, du \]This simplifies to: \[ \frac{1}{2} \int_{0}^{4} f(u) \, du \]

Step 3: Apply the Given Integral Value

From the problem, we know that \( \int_{0}^{4} f(x) \, dx = 10 \). Since we transformed the integral to the form \( \int_{0}^{4} f(u) \, du \), which is equivalent to \( \int_{0}^{4} f(x) \, dx \), we substitute the value:\[ \frac{1}{2} \times 10 = 5 \].

Key Concepts

Integration Techniques

Integration is a core component of calculus, widely used to find areas under curves, among other applications. There are various techniques available to perform integration, each suitable for different types of functions. When working with integrals, selecting the right technique is crucial for simplifying and solving the problem successfully.

• Substitution: Often used when an integral contains a composition of functions. The substitution technique involves changing variables to simplify the integration process.
• Integration by Parts: Used when an integral is the product of two functions. It stems from the product rule of differentiation and helps break down complex integrals.
• Partial Fractions: Effective for rational functions. It involves decomposing a complex rational expression into simpler fractions that are easier to integrate.

In our exercise, we utilized the substitution technique, allowing us to transform a more complex integral into a simpler form.

Variable Substitution

Variable substitution is a powerful technique to simplify integrals by changing the variable of integration. It is particularly useful when dealing with functions within other functions, or when a differential change is needed.When using substitution, you should:

• Identify a part of the integral to substitute with a new variable, often denoted by \( u \).
• Determine the relationship between \( dx \) and \( du \). In our problem, \( u = 2x \) led to \( du = 2 \, dx \), hence \( dx = \frac{1}{2} du \).
• Adjust the limits of integration based on the new variable. Originally, \( x = 0 \) and \( x = 2 \) transform to \( u = 0 \) and \( u = 4 \).

Through proper substitution, the integral becomes \[ \int_{0}^{4} f(u) \left(\frac{1}{2}\right) \, du \].This transformation made it possible to apply the given integral condition efficiently.

Definite Integrals

Definite integrals are used to calculate the accumulated area under a curve over a specified interval. They give definite numerical results, unlike indefinite integrals that represent a family of functions with an arbitrary constant.In the context of definite integrals:

• The integration bounds (limits) determine the range over which the function is integrated. It is essential to correctly adjust these bounds during substitution.
• The result provides a specific value, representing the total accumulation of a quantity, such as area, over the given interval.
• In our exercise, the transformed integral \[ \frac{1}{2} \int_{0}^{4} f(u) \, du \] utilizes the given information that \( \int_{0}^{4} f(x) \, dx = 10 \). Thus, the definite integral gives us precisely \( \frac{1}{2} \times 10 = 5 \), reflecting the portion of the area over the interval \([0, 4]\).

This demonstrates the practical application of definite integrals in solving real-world problems and understanding the behavior of functions over intervals.