Problem 21 - Calculus: Early Transcendentals

Question

Evaluate the integral. \( \displaystyle \int^3_{-2} (x^2 - 3) \,dx \)

Step-by-Step Solution

Answer

The value of the integral is \(-\frac{10}{3}\).

Step 1: Identify the Integral

Given the integral is \( \int_{-2}^{3} (x^2 - 3) \, dx \). We want to evaluate this definite integral over the interval \([-2, 3]\).

Step 2: Find the Antiderivative

To solve the integral, first find the antiderivative of the integrand \(x^2 - 3\). The antiderivative of \(x^2\) is \(\frac{x^3}{3}\), and the antiderivative of \(-3\) is \(-3x\). Thus, the antiderivative of \(x^2 - 3\) is \(\frac{x^3}{3} - 3x\).

Step 3: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that to evaluate \(\int_{a}^{b} f(x) \, dx\), you find the antiderivative \(F(x)\) and compute \(F(b) - F(a)\). Here, compute \(F(3) - F(-2)\), where \(F(x) = \frac{x^3}{3} - 3x\).

Step 4: Evaluate the Antiderivative at the Upper Limit

Calculate \(F(3) = \frac{3^3}{3} - 3(3)\). Simplify this to \(F(3) = 9 - 9 = 0\).

Step 5: Evaluate the Antiderivative at the Lower Limit

Calculate \(F(-2) = \frac{(-2)^3}{3} - 3(-2)\). Simplify this to \(F(-2) = \frac{-8}{3} + 6\). Convert 6 to a fraction to get \(\frac{-8}{3} + \frac{18}{3} = \frac{10}{3}\).

Step 6: Compute the Definite Integral

Using the values obtained, calculate \(F(3) - F(-2) = 0 - \frac{10}{3} = -\frac{10}{3}\). Thus, the value of the definite integral is \(-\frac{10}{3}\).

Key Concepts

Antiderivative

The concept of an antiderivative is fundamental when evaluating integrals. In simple terms, the antiderivative of a function is a function whose derivative is the original function we started with.

When dealing with a definite integral, like the one given in the exercise, we need to find the antiderivative of the integrand, which is the function to be integrated. Here, the integrand is expressed as a polynomial: \(x^2 - 3\).

To find the antiderivative of each term:

• For \(x^2\): The antiderivative is \(\frac{x^3}{3}\). This is because the power of \(x\) is increased by 1, making it \(x^3\), and we divide by this new power.
• For \(-3\): The antiderivative is \(-3x\), as it is equivalent to the derivative rule applied reverse; the constant \(3\) is multiplied directly by \(x\).

Combining these results, the antiderivative of \(x^2 - 3\) is \(\frac{x^3}{3} - 3x\). This function will be crucial for evaluating the definite integral.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) bridges the concepts of differentiation and integration. It is composed of two main parts, but in evaluating definite integrals, we focus on its second part.

This part states that if \(F(x)\) is the antiderivative of a function \(f(x)\), then the definite integral of \(f(x)\) from \(a\) to \(b\) is given by \(F(b) - F(a)\).

Applying the FTC to Our Problem

With our given integral \(\int^3_{-2} (x^2 - 3) \,dx\), we first found the antiderivative: \(F(x) = \frac{x^3}{3} - 3x\). According to the FTC, to evaluate this definite integral:

• First, compute \(F(b)\), which is \(F(3)\).
• Then, compute \(F(a)\), which in our case is \(F(-2)\).
• Finally, subtract \(F(a)\) from \(F(b)\) to get: \(F(3) - F(-2)\).

This calculation gives us the area under the curve of \(x^2 - 3\) from \(-2\) to \(3\), resulting in \(-\frac{10}{3}\). This outcome shows the integral of a function can yield negative or positive values depending on the function's position relative to the x-axis within the interval.

Polynomial Integration

Polynomial integration involves integrating expressions consisting of sums of variable powers multiplied by coefficients. The rules for integrating polynomials are straightforward and derived from the power rule of differentiation, but applied in reverse.

Steps for Polynomial Integration

• Increase the Power: For a term \(x^n\), increase the power by 1 to get \(x^{n+1}\).
• Divide by the New Power: Divide the term by the new power, resulting in \(\frac{x^{n+1}}{n+1}\).
• Combine Constants: For constant terms, simply multiply by \(x\) since the derivative of a constant is simply the constant itself with respect to \(x\).

In our exercise, the polynomial \(x^2 - 3\) involves:

• \(x^2\): Becoming \(\frac{x^3}{3}\).
• \(-3\): Transforming to \(-3x\).

Thus, the integrated antiderivative \(\frac{x^3}{3} - 3x\) encompasses all the polynomial powers and constants, providing a full picture for evaluation when combined with limits of integration.