Problem 5 - Calculus

Question

Evaluate the indicated integrals. $$ \int_{2}^{8} z\left(2 z^{2}-3\right)^{1 / 3} d z $$

Step-by-Step Solution

Answer

The evaluated integral is approximately 115.34.

Step 1: Identify the Type of Integral

The integral \( \int_{2}^{8} z(2z^2 - 3)^{1/3} \, dz \) is a standard integral that involves a composite function \((2z^2 - 3)^{1/3}\). Let's consider substitution to simplify the expression for integration.

Step 2: Choose an Appropriate Substitution

Let's use substitution to simplify the integration process. Set \( u = 2z^2 - 3 \). Then \( \frac{du}{dz} = 4z \), which simplifies to \( dz = \frac{du}{4z} \). Substitute back into the integral.

Step 3: Substitute and Simplify

Substitute \( u = 2z^2 - 3 \) into the integral and replace \( dz \):\[ \int z u^{1/3} \cdot \frac{du}{4z} = \frac{1}{4} \int u^{1/3} \, du \]The \( z \) terms cancel out, simplifying the integral greatly.

Step 4: Evaluate the Simplified Integral

Now evaluate the integral \( \frac{1}{4} \int u^{1/3} \, du \). This results in:\[ \frac{1}{4} \cdot \frac{3}{4} u^{4/3} = \frac{3}{16} u^{4/3} + C \]

Step 5: Substitute Back for z

Re-substitute \( u = 2z^2 - 3 \) back into the expression to revert to the variable \( z \):\[ \frac{3}{16} (2z^2 - 3)^{4/3} \]

Step 6: Apply the Limits of Integration

Applying the limits of integration (from 2 to 8):\[ \frac{3}{16} [(2(8)^2 - 3)^{4/3} - (2(2)^2 - 3)^{4/3}] \]Calculating the expressions in the brackets:\( 2(8)^2 - 3 = 125 \) and \( 2(2)^2 - 3 = 5 \).Then compute:\[ \frac{3}{16} [125^{4/3} - 5^{4/3}] \]

Step 7: Calculate Final Numerical Values

Compute each term:\( 125^{4/3} = 625 \, (since \, 125^{1/3} = 5) \) and \( 5^{4/3} \approx 11.18034 \).Now calculate:\[ \frac{3}{16} [625 - 11.18034] \approx \frac{3}{16} \times 613.81966 \approx 115.34 \]

Key Concepts

Integration by Substitution

Integration by substitution is a powerful technique used to simplify the integration of complex functions. This method is particularly useful when dealing with an integral that is difficult to solve directly. By making an appropriate substitution, we can often transform the integral into a simpler form.

Here's how it works:

• Identify a part of the integral that can be replaced with a new variable. This usually involves the inner part of a composite function.
• Express the differential of the new variable in terms of the original variable to facilitate substitution.
• Replace all instances of the identified expression in the integral with the new variable and its differential.
• After computing the integral in terms of the new variable, substitute back the original variable expressions.

For example, in the integral \( \int z(2z^2 - 3)^{1/3} \, dz \), we noticed that the expression \(2z^2 - 3\) could be replaced by \(u\). This substitution simplifies the integral since the \(z\) terms cancel out, leaving a much simpler integral of \( \frac{1}{4} \int u^{1/3} \, du \). The result is then adjusted back to the original variable, allowing us to evaluate the integral with ease once again.

Definite Integral

A definite integral represents the area under the curve of a function between two specific limits, providing a numerical value. This is opposed to an indefinite integral, which includes a constant of integration and results in a family of functions. Definite integrals have broad applications, including computing areas, volumes, and even in solving differential equations.

When solving a definite integral:

• Determine the function to be integrated and identify the limits of integration.
• Apply the necessary integration method, such as substitution, to simplify the integral, if needed.
• Evaluate the integral and then apply the limits of integration by finding the difference in the function values at the upper and lower bounds.

In our exercise, the integral \( \int_{2}^{8} z(2z^2 - 3)^{1/3} \, dz \) is a definite integral with limits from 2 to 8. After simplifying and integrating using substitution, we substitute the limits into the evaluated integral to find the area under the function curve between these limits.

This process results in a specific numerical value, \(115.34\), representing this area, highlighting the utility of definite integrals in computations and real-world applications.

Composite Function

A composite function is a function made up of two or more functions, where one function is applied within the other. This concept is pivotal in understanding many calculus problems as it often shows up in integration and differentiation tasks.

The composite function \((2z^2 - 3)^{1/3}\) presents a classic example. Here, the inside function is \(2z^2 - 3\) and the outside function is raising to the power \(1/3\). When integrating a composite function, it is often useful to utilize the substitution method, as this can simplify the overall integration.

Handling composite functions involves:

• Identifying the inside and outside functions.
• Using substitution to simplify the expression, focusing on the inside function usually helps to manage the complexity.
• Simplifying the integral to a more straightforward form for calculation.

In our original exercise, substituting \(u = 2z^2 - 3\) helps get a grip on the composite function structure and manage its integration more effectively. Understanding how to break down these functions is crucial, making intricate calculus problems more manageable.