Problem 14 - Applied Calculus

Question

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{2}^{5}\left(x^{3}-\pi x^{2}\right) d x $$

Step-by-Step Solution

Answer

The exact value of the integral is \( \frac{609}{4} - \frac{117\pi}{3} \).

Step 1: Identify the Function to Integrate

The function we need to integrate is given by \( f(x) = x^3 - \pi x^2 \). This is a polynomial function, which makes it straightforward to integrate using basic calculus rules.

Step 2: Find the Antiderivative

To find the antiderivative of \( f(x) = x^3 - \pi x^2 \), we integrate each term separately. The antiderivative of \( x^3 \) is \( \frac{x^4}{4} \), and the antiderivative of \( \pi x^2 \) is \( \frac{\pi x^3}{3} \). Thus, the antiderivative \( F(x) \) is \( F(x) = \frac{x^4}{4} - \frac{\pi x^3}{3} + C \), where \( C \) is a constant of integration.

Step 3: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, the definite integral from \( a \) to \( b \) of \( f(x) \) can be found by \( F(b) - F(a) \), where \( F(x) \) is the antiderivative of \( f(x) \). In this case, we need to compute \( F(5) - F(2) \).

Step 4: Evaluate the Antiderivative at the Bounds

Compute \( F(5) = \left(\frac{5^4}{4} - \frac{\pi \cdot 5^3}{3}\right) \) and \( F(2) = \left(\frac{2^4}{4} - \frac{\pi \cdot 2^3}{3}\right) \). These values are \( F(5) = \frac{625}{4} - \frac{125\pi}{3} \) and \( F(2) = \frac{16}{4} - \frac{8\pi}{3} \), which simplify to \( F(2) = 4 - \frac{8\pi}{3} \).

Step 5: Calculate the Definite Integral

Subtract \( F(2) \) from \( F(5) \): \[F(5) - F(2) = \left( \frac{625}{4} - \frac{125\pi}{3} \right) - \left( 4 - \frac{8\pi}{3} \right).\] Simplify this expression:\[= \frac{625}{4} - 4 - \frac{125\pi}{3} + \frac{8\pi}{3} \= \frac{625}{4} - 4 - \frac{117\pi}{3}.\] Further simplifying, we find:\[\frac{625}{4} - 4 = \frac{609}{4},\] thus resulting in \[\frac{609}{4} - \frac{117\pi}{3}.\]

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a cornerstone in understanding how derivatives and integrals are related. It comprises two parts that bridge the concept of derivative of a function and its integral. The first part essentially connects the process of differentiation and integration, while the second part provides a method to evaluate definite integrals.

In the context of definite integrals, as seen in the exercise, the second part of the theorem asserts that if you have a continuous function on a closed interval \(a, b\), then the integral from \(a\) to \(b\) of the function can be calculated using its antiderivative. Essentially, if \(F(x)\) is an antiderivative of \(f(x)\), then:

• \(\int_{a}^{b}f(x)dx = F(b) - F(a)\)

In the given problem, we applied this property to find the definite integral by first finding an antiderivative of the polynomial function and then computing \(F(5) - F(2)\). This powerful theorem simplifies the calculation of areas and accumulated quantities tremendously.

Antiderivative

An antiderivative of a function is another function whose derivative yields the original function. It's like reverse engineering differentiation. In the exercise, the function \(f(x) = x^3 - \pi x^2\) needed to be integrated to find its antiderivative.

The process of finding antiderivatives is called integration, and it involves certain rules that simplify this process for different types of functions. For polynomials, each term is treated individually by applying the reverse power rule: \(\frac{x^n}{n}\) becomes the antiderivative of \(x^{n-1}\). So for \(x^3\):

• Antiderivative = \(\frac{x^4}{4}\)

Similarly, for \(\pi x^2\):

• Antiderivative = \(\frac{\pi x^3}{3}\)

Thus, the complete antiderivative of the function becomes \( F(x) = \frac{x^4}{4} - \frac{\pi x^3}{3} + C\), where \(C\) is an arbitrary constant that does not affect the definite integral as it cancels out when computing the difference \(F(b) - F(a)\).

Polynomial Integration

Polynomial integration is a process to find the area under the curve of a polynomial function. These types of functions are expressions involving sums of powers of \(x\) with different coefficients, like \(x^3 - \pi x^2\) in the exercise.

Integrating polynomials is straightforward because each term can be handled separately using the power rule for integration. This rule states that to integrate \(x^n\), simply add 1 to the exponent and divide by the new exponent: \(x^n \rightarrow \frac{x^{n+1}}{n+1}\). Since each polynomial term can be handled independently, this makes the integration process faster and efficient.

In the exercise, the polynomial \(x^3 - \pi x^2\) was split into two terms: \(x^3\) and \(-\pi x^2\), and each was integrated separately. This method provides the antiderivative, which is crucial for evaluating the definite integral using the Fundamental Theorem of Calculus. Understanding this concept is essential for handling complex polynomial functions in calculus.