91Ó°ÊÓ

The concept of a definite integral is foundational in calculus, serving as a tool for calculating the area under a curve over a specific interval. In our exercise, to find the average value of the function \( f(z) = 4z - 3z^2 \) over the interval \([-2, 2]\), we first need to calculate the definite integral of the function across this closed interval.
The definite integral \( \int_{-2}^{2} (4z - 3z^2) \, dz \) provides a single numerical value representing the net area enclosed by the curve \( 4z - 3z^2 \) and the \( z \)-axis, between the limits \( z = -2 \) and \( z = 2 \).

• The definite integral is a "summation" over a continuous range, and it accounts for both positive and negative contributions to the area, based on the curve's position relative to the \( z \)-axis.
• It's calculated using the antiderivative (also known as an indefinite integral) evaluated at these boundary points—the starting and ending values \(-2\) and \(2\).

With this integral, we encapsulate how much space the function occupies over the given interval, essential for solving many real-world problems.

Integration, the reverse process of differentiation, involves finding a function's antiderivative. In this context, we employ basic integration techniques to dissect the function \( f(z) = 4z - 3z^2 \). Here's a breakdown of this process:
First, we break down the expression to integrate each component separately:

• The integral of \(4z\) is calculated as \(2z^2\) because the power rule for integration states that \( \int z^n \, dz = \frac{z^{n+1}}{n+1} \).
• The integral of \(-3z^2\) becomes \(-z^3\) following the same principle, consistent with the formula \( \int az^n \, dz = \frac{a}{n+1} z^{n+1} \).

Thus, the antiderivative is \( 2z^2 - z^3 + C \), with \( C \) denoting the constant of integration.By using these steps, students can manually compute integrals for polynomials, a crucial skill. Mastery of integration techniques allows for evaluating more complex functions and solving various physics and engineering problems.

Mathematical problem-solving requires a structured approach, as seen in calculating the average value of a function across an interval. In this exercise, we apply a methodical, step-by-step process:

• **Understand the Problem**: Identify what you're solving for, here, the average value across \([-2, 2]\).
• **Establish the Mathematical Procedures**: Set up and simplify the definite integral \( \int_{-2}^{2} (4z - 3z^2) \, dz \).
• **Apply Calculations**: Evaluate the antiderivative at the boundaries, \( z = 2 \) and \( z = -2 \), to ascertain the total integral value.
• **Compute the Average Value**: Finally, divide the result by the interval's length using the formula \( \frac{1}{b-a} \int_{a}^{b} f(z) \, dz \), resulting in \(-\frac{8}{3}\).

These steps illustrate the application of calculus in deriving practical answers, fostering analytical thinking. Developing such problem-solving strategies enhances understanding and supports tackling a range of mathematical questions.