Problem 37 - Calculus and its applications

Question

Evaluate each integral. Then state whether the result indicates that there is more area above or below the \(x\) -axis or that the areas above and below the axis are equal. $$ \int_{-1}^{1}\left(x^{3}-3 x\right) d x $$

Step-by-Step Solution

Answer

The integral evaluates to 0, meaning the areas above and below the x-axis are equal.

Step 1: Understanding the Function

The given function is \( f(x) = x^3 - 3x \). This is a cubic polynomial, which can have sections where it is above or below the x-axis. Our goal is to evaluate the definite integral from \(-1\) to \(1\) and determine if there is more area above or below the x-axis.

Step 2: Set Up the Integral

We want to compute the definite integral: \[ \int_{-1}^{1} (x^3 - 3x) \, dx \]. This will involve finding an antiderivative of the function \( x^3 - 3x \).

Step 3: Find the Antiderivative

The antiderivative of \( x^3 \) is \( \frac{x^4}{4} \) and the antiderivative of \(-3x\) is \(-\frac{3x^2}{2} \). Therefore, an antiderivative of \( x^3 - 3x \) is \( F(x) = \frac{x^4}{4} - \frac{3x^2}{2} \).

Step 4: Evaluate the Definite Integral

Use the Fundamental Theorem of Calculus to evaluate \[ \left[ \frac{x^4}{4} - \frac{3x^2}{2} \right]_{-1}^{1} = \left( \frac{1^4}{4} - \frac{3 \cdot 1^2}{2} \right) - \left( \frac{(-1)^4}{4} - \frac{3 \cdot (-1)^2}{2} \right). \] Simplifying each term gives \( \frac{1}{4} - \frac{3}{2} - (\frac{1}{4} - \frac{3}{2}) \).

Step 5: Simplify the Result

The values \( \left( \frac{1}{4} - \frac{3}{2} \right) - \left( \frac{1}{4} - \frac{3}{2} \right) \) give \( -\frac{5}{4} - (-\frac{5}{4}) = 0 \). This implies that the areas above and below the x-axis within the interval are equal.

Key Concepts

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, which means its highest power of the variable is three. In general, a cubic polynomial is expressed in the form \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \text{ and } d \) are constants and \( a eq 0 \). These types of polynomials have unique characteristics that define their behavior.

• Shape: The graph of a cubic polynomial is an S-shaped curve, with potential for one or two turning points where it can go from increasing to decreasing or vice versa.
• Roots: A cubic polynomial might have up to three real roots, corresponding to the x-intercepts of the graph.
• Sign changes: The function may cross the x-axis multiple times, resulting in regions above and below the axis.

Understanding these properties helps in evaluating integrals of cubic polynomials, as they indicate where the function may change direction and have areas calculated both positive and negative.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a central theorem that links differentiation and integration, two main concepts in calculus. It consists of two parts that provide a powerful way to evaluate definite integrals.

• Part 1: This part states that the integral of a function gives the accumulation of the area under the curve and is related to its antiderivative. If \( F \) is an antiderivative of \( f \) on an interval, then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).
• Part 2: It suggests that if \( f \) is continuous on \([a, b]\), then its antiderivative \( F \) is also continuous, and the rate of change of the accumulation function is the original function itself.

In practical terms, the Fundamental Theorem of Calculus allows us to evaluate definite integrals by finding an antiderivative of the function. This theorem simplifies the process of finding the area under curves by giving a straightforward formula for the evaluation.

Antiderivative

An antiderivative, also known as an indefinite integral, of a function is a new function whose derivative is the original function. Finding an antiderivative is the reverse process of differentiation.

• Basic Concept: If \( F(x) \) is an antiderivative of \( f(x) \), then \( F'(x) = f(x) \). This means differentiating \( F(x) \) will return \( f(x) \).
• General Form: The most general form of an antiderivative includes a constant \( C \), written as \( F(x) + C \), because the derivative of a constant is zero.
• Finding Antiderivatives: The process of finding an antiderivative involves rules such as power rule: the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \) if \( n eq -1 \).

In the exercise provided, the antiderivatives were found for both components of the cubic polynomial \(x^3\) and \(-3x\). This was essential for applying the Fundamental Theorem of Calculus to determine the net area under the curve between the specified limits.