Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the definite integral. $$\int_{0}^{2} 8 x^{3} d x$$

Step-by-Step Solution

Answer

The value of the definite integral \(\int_{0}^{2} 8x^{3} dx\) is 32.

Step 1: Find the antiderivative of the function

To find the antiderivative of the function $$8x^{3}$$, we will use the power rule for integration, which states that $$\int x^n dx = \dfrac{x^{n+1}}{n+1} + C$$ where $$n$$ is a constant and $$C$$ is the constant of integration. Applying the power rule to our function, we get: $$\int 8x^{3} dx = \dfrac{8x^{4}}{4} + C = 2x^{4} + C$$

Step 2: Apply the Fundamental Theorem of Calculus

Now that we have the antiderivative, we can apply the Fundamental Theorem of Calculus to evaluate the definite integral. The theorem states that: $$\int_{a}^{b} f(x) dx = F(b) - F(a)$$ where $$F(x)$$ is the antiderivative of the function $$f(x)$$ and $$[a, b]$$ is the interval of integration. Using our antiderivative from Step 1 and the given interval $$[0, 2]$$, we obtain: $$\int_{0}^{2} 8x^{3} dx = 2x^{4}\Big|_{0}^{2} = 2(2)^{4} - 2(0)^{4}$$

Step 3: Calculate the result

Finally, we compute the result: $$2(2)^{4} - 2(0)^{4} = 2(16) = 32$$ Thus, the value of the definite integral is 32: $$\int_{0}^{2} 8x^{3} dx = 32$$

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental technique used to find the antiderivative of polynomial functions. It simplifies the process of integration, especially with terms composed of a variable raised to a power. The general formula is:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Here, \( n \) represents a constant exponent, and \( C \) is the constant of integration.
This rule is essential in transforming derivatives back into their original functions—a process known as finding the antiderivative.
When applying the power rule:

• Increase the exponent by 1.
• Divide the term by the new exponent.

For example, in the integral \( \int 8x^3 \, dx \), increasing the exponent of 3 by 1 gives \( x^4 \), and dividing by 4 results in \( 2x^4 \). This gives the antiderivative necessary for further calculations in definite integrals.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, two major concepts in calculus. It essentially states that integration and differentiation are inverse processes.
The theorem comes in two parts, but for solving definite integrals, we focus on the second part, which states:

• If \( F(x) \) is the antiderivative of \( f(x) \) on the interval \([a, b]\), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).

This formula allows us to evaluate a definite integral by calculating the difference between the antiderivative evaluated at the upper limit and the lower limit of the integral.
In our exercise, we found the antiderivative of function \( 8x^3 \) as \( 2x^4 \), and applied the theorem with limits 0 to 2. This led to:
\( 2(2)^4 - 2(0)^4 = 32 \). This result gives us the exact area under the curve of the function from \( x = 0 \) to \( x = 2 \).

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative yields the original function. When you take the antiderivative of a function, you essentially find a function \( F(x) \) such that \( F'(x) = f(x) \).
Antiderivatives are crucial in solving integrals, particularly definite ones, as they allow us to calculate the area under curves with accuracy. Every antiderivative comes with a constant term, \( C \), because the derivative of a constant is zero. Thus, there are infinitely many antiderivatives differing only by a constant.
In the exercise, the antiderivative of \( 8x^3 \) is found using the power rule as \( 2x^4 + C \) (where \( C \) is generally not needed for definite integrals). Once the antiderivative is determined, it is used in conjunction with the Fundamental Theorem of Calculus to evaluate definite integrals effectively.