Problem 38 - Calculus Early Transcendentals

Question

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{4} x(x-2)(x-4) d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral of the function x(x-2)(x-4) with respect to x from 0 to 4. Answer: The integral of x(x-2)(x-4) with respect to x from 0 to 4 is equal to 0.

Step 1: Find the antiderivative of x(x-2)(x-4)

To find the antiderivative of this function, we first have to expand the polynomial and then find the indefinite integral term by term. Expanding, we have $$x(x-2)(x-4) = x(x^2 - 6x + 8) = x^3 - 6x^2 + 8x.$$ Now, we can find the antiderivative of this polynomial term by term. Recall that the antiderivative of x^n is (x^(n+1))/(n+1). Thus, we have: $$\int (x^3 - 6x^2 + 8x) d x = \frac{1}{4}x^4 - 2x^3 + 4x^2 + C.$$

Step 2: Evaluate the antiderivative at the limits of integration and subtract

Now, we will apply the Fundamental Theorem of Calculus to evaluate the definite integral: $$\int_{0}^{4} x(x-2)(x-4) dx = \left[\frac{1}{4}x^4 - 2x^3 + 4x^2\right]_0^4$$ We will first plug in 4 into the antiderivative: $$\frac{1}{4}(4)^4 - 2(4)^3 + 4(4)^2 = 64 - 128 + 64 = 0.$$ Now, let's plug in 0: $$\frac{1}{4}(0)^4 - 2(0)^3 + 4(0)^2 = 0.$$ Therefore, the definite integral is equal to: $$\int_{0}^{4} x(x-2)(x-4) dx = 0 - 0 = 0.$$

Key Concepts

Polynomial Integration

Polynomial integration involves finding the integral, or antiderivative, of a polynomial function. A polynomial is an expression composed of variables and coefficients, with non-negative integer exponents. To integrate a polynomial, we apply the process of finding the indefinite integral term by term.

Here's a step-by-step look at how polynomial integration works:

• Break down the polynomial into individual terms, focusing on powers of the variable.
• Apply the power rule for integration to each term separately. This rule states that the antiderivative of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\), where \(n\) is not equal to -1.
• Add a constant of integration \(C\) to indicate the indefinite nature of the antiderivative.

In our original exercise, we started by expanding the polynomial \(x(x-2)(x-4)\) into \(x^3 - 6x^2 + 8x\). Then, we integrated each term using the power rule:

• \(\int x^3 \, dx = \frac{1}{4}x^4\)
• \(\int -6x^2 \, dx = -2x^3\)
• \(\int 8x \, dx = 4x^2\)

Adding these gives the complete antiderivative: \(\frac{1}{4}x^4 - 2x^3 + 4x^2 + C\). This result is crucial for solving definite integrals efficiently.

Definite Integral

The definite integral is a type of integral that computes the net area under a curve between two specified limits, usually on the x-axis. It provides valuable information about the total accumulation of quantities, like area, over an interval. In our exercise, we calculate the definite integral of the polynomial \(x(x-2)(x-4)\) over the interval from \(0\) to \(4\).

The Fundamental Theorem of Calculus connects differentiation and integration, making it possible to evaluate definite integrals directly:

• We first find the antiderivative of the function, which is the same as its indefinite integral.
• Next, we evaluate this antiderivative at the upper limit and lower limit of the integral.
• Lastly, we subtract the lower limit value from the upper limit value to find the net area.

In the given problem, after finding the antiderivative \(\frac{1}{4}x^4 - 2x^3 + 4x^2\), we assess it at \(x = 4\) yielding \(0\), and also at \(x = 0\), which results in \(0\). The result of the definite integral is \(0\), meaning the graph above and below the x-axis during the interval from \(0\) to \(4\) perfectly balances to nothing.

Antiderivative

An antiderivative is an expression that represents all the functions that could have been differentiated to result in a given function. In simpler terms, the antiderivative is the reverse of differentiation.

Finding an antiderivative is the central task in integration, especially when solving a definite integral.

• The antiderivative of a function \(f(x)\) is denoted as \(F(x)\) such that \(F'(x) = f(x)\).
• The process of determining the antiderivative is sometimes called "indefinite integration."
• A common formula involved is \(\int x^n \,dx = \frac{x^{n+1}}{n+1} + C\).

In our original problem, we needed to identify the antiderivative of the expanded polynomial \(x^3 - 6x^2 + 8x\) to solve the integral. Each term in the polynomial was integrated separately to find this antiderivative.
This concept is crucial as it allows the application of the Fundamental Theorem of Calculus to evaluate definite integrals efficiently. Understanding antiderivatives paves the way for mastering more complex calculus topics.