Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the definite integral. $$\int_{0}^{1}\left(x^{3}-2 x^{2}+1\right) d x$$

Step-by-Step Solution

Answer

The value of the definite integral \(\int_{0}^{1}(x^{3}-2 x^{2}+1) d x\) is \(\frac{7}{12}\).

Step 1: Find the antiderivative of the function

To find the antiderivative of \(x^3 - 2x^2 + 1\), we will apply the power rule of integration separately to each term. The power rule states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(n\) is a constant and \(C\) is the constant of integration. Applying the power rule to each term in the function, we get: \[\int (x^3 - 2x^2 + 1) dx = \frac{x^4}{4} - 2\frac{x^3}{3} + x + C\] Since we are evaluating a definite integral, we won't need the constant of integration, so the antiderivative we will use is: \[F(x) = \frac{x^4}{4} - \frac{2x^3}{3} + x\]

Step 2: Apply the Fundamental Theorem of Calculus

Now we will 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 \(f(x)\). In our case, \(a = 0, b = 1\), and \(F(x) = \frac{x^4}{4} - \frac{2x^3}{3} + x\). Now, let's plug in the limits of integration: \[\int_{0}^{1}\left(x^{3} - 2x^{2} + 1\right) dx = F(1) - F(0) = \left[\frac{1^4}{4} - \frac{2\cdot 1^3}{3} + 1\right] - \left[\frac{0^4}{4} - \frac{2\cdot 0^3}{3} + 0\right]\]

Step 3: Simplify and compute the result

Let's now simplify the expression and compute the result: \[\int_{0}^{1}\left(x^{3}-2 x^{2}+1\right) d x = \left[\frac{1}{4} - \frac{2}{3} + 1\right] - \left[0 - 0 + 0\right] = \frac{1}{4} - \frac{2}{3} + 1\] Now, let's find a common denominator for the fractions and combine them: \[= \frac{3}{12} - \frac{8}{12} + \frac{12}{12} = \frac{3 - 8 + 12}{12} = \frac{7}{12}\] So, \[\int_{0}^{1}\left(x^{3}-2 x^{2}+1\right) d x = \frac{7}{12}.\] Thus, the value of the definite integral is \(\frac{7}{12}\).

Key Concepts

Antiderivative Calculation

Understanding the calculation of an antiderivative is fundamental in tackling integration problems. An antiderivative, also known as an indefinite integral, refers to the reverse process of differentiating a function. It represents a family of functions that, when differentiated, give the original function. To find an antiderivative, you apply the reverse operations of differentiation.

For example, consider the function given by the term \(x^3 - 2x^2 + 1\). To find its antiderivative, we reverse the power rule for derivatives. For the term \(x^3\), we add 1 to the exponent and then divide by the new exponent, resulting in \(\frac{x^4}{4}\). Repeat the process for each term, remembering to include a constant of integration \(C\) for indefinite integrals. However, when dealing with definite integrals, the constant of integration is unnecessary, as it will cancel out when evaluating the limits of integration.

Power Rule for Integration

Integration and differentiation are essential concepts in calculus, often acting as inverse processes. The power rule for integration is a straightforward technique to integrate polynomial functions. It states that for any real number n, different from -1, the integral of \(x^n\) with respect to x is given by \(\frac{x^{n+1}}{n+1}\) plus the constant of integration \(C\).

To apply the rule:

• Increment the exponent of \(x\) by one.
• Divide the term by this new exponent value.

The power rule greatly simplifies the integration process for polynomials like \(x^3\) or \(x^2\), transforming the integral into easily manageable expressions. Always remember that the rule does not apply when \(n = -1\), as this leads to the natural logarithm function, not a power function.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration, providing a way to evaluate definite integrals. In its essence, the theorem states that if a function \(f(x)\) is continuous on the interval \[a, b\], and if \(F(x)\) is an antiderivative of \(f(x)\) on the interval, then the definite integral of \(f(x)\) from \(a\) to \(b\) is given by \(F(b) - F(a)\).

This process consists of two main steps:

• Find an antiderivative of the function you are integrating.
• Evaluate this antiderivative at the upper and lower bounds of the interval, then take the difference.

As seen in our example, after finding the antiderivative \(F(x)\), we used the bounds 0 and 1 to find \(F(1) - F(0)\), which gives the value of the integral \(\frac{7}{12}\). The Fundamental Theorem of Calculus is a powerful and widely used tool that makes solving definite integrals much more manageable.