Problem 26 - Applied Calculus

Question

Estimate \(\int_{0}^{10} f(x) g^{\prime}(x) d x\) if \(f(x)=x^{2}\) and \(g\) has the values in the following table. $$ \begin{array}{c|c|c|c|c|c|c} \hline x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline g(x) & 2.3 & 3.1 & 4.1 & 5.5 & 5.9 & 6.1 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The estimated value of the integral is 282.4.

Step 1: Understand the Integration by Parts Formula

We use integration by parts to solve \( \int_{0}^{10} f(x) g^{\prime}(x) dx \). The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \). We can choose \( u = f(x) \) and \( dv = g'(x) \, dx \).

Step 2: Identify Variables for Integration by Parts

Set \( u = x^2 \) so that \( du = 2x \, dx \). Since \( dv = g'(x) \, dx \), by the Fundamental Theorem of Calculus, we have \( v = g(x) \).

Step 3: Compute Initial Evaluation \( uv \)

Evaluate \( uv \) at the boundaries: \( [u(10)g(10) - u(0)g(0)] = [10^2 \times 6.1 - 0^2 \times 2.3] = 610 \).

Step 4: Setup Integral of \( v du \)

We need \( \int v \, du = \int g(x) \times 2x \, dx \) over the interval [0, 10]. This integral requires approximate numerical methods, because exact values of \( g(x) \) are only given at specific points.

Step 5: Use Trapezoidal Rule to Approximate \( \int g(x) \, 2x \, dx \)

Approximate \( \int_{0}^{10} g(x) \, 2x \, dx \) using the trapezoidal rule: \( \text{Trapezoidal Rule: } \frac{h}{2} [f(a) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n−1}) + f(b)] \).Here, the partition intervals are \( (0, 2, 4, 6, 8, 10) \), \( h = 2 \), and \( f(x) = 2x \cdot g(x) \). \[ \int_{0}^{10} g(x) \, 2x \, dx = \frac{2}{2} (2 \times 0 \times 2.3 + 2(2 \times 3.1) + 2(4 \times 4.1) + 2(6 \times 5.5) + 2(8 \times 5.9) + 10 \times 6.1) \].

Step 6: Calculate Using Trapezoidal Approximations

Plug each value into the trapezoidal formula: \( = 0 + 12.4 + 32.8 + 66.0 + 94.4 + 122.0 = 327.6 \).

Step 7: Combine Parts from Integration by Parts

Substitute back into the integration by parts formula. The total estimate is \( 610 - 327.6 = 282.4 \).

Key Concepts

Trapezoidal Rule

The trapezoidal rule is a handy method for estimating the definite integral of a function when we don't know its exact expression but have values at certain points. This approach approximates the region under a curve as a series of trapezoids, which is why it's called the trapezoidal rule.
The formula used to approximate the integral is given by:

• \[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n) \right] \]

Here, \( h \) is the width of each subinterval, \( f(x_i) \) is the function value at each point, and \( [a, b] \) is the interval over which we're integrating.
This method is particularly useful when dealing with data available at discrete points, as it can produce a good approximation without complex calculations. It provides a simple yet effective way to "connect the dots" and estimate areas when exact values aren't accessible.

Approximate Numerical Methods

When exact solutions to integrals or other mathematical problems are challenging to obtain, we turn to approximate numerical methods. These methods allow us to estimate solutions with reasonable accuracy without deriving complex formulas.
Some common numerical methods include:

• Trapezoidal Rule: Approximates the integral by dividing the area under a curve into trapezoids.
• Simpson's Rule: Uses parabolic arcs instead of straight lines for a more accurate estimate.
• Monte Carlo Method: Employs random sampling to approximate the value of a complex integral.

In the context of the problem, the trapezoidal rule is used to estimate the integral since the exact function is only defined at discrete points. These numerical methods are crucial in various fields, such as engineering and physics, where they help in simulating and modeling real-world scenarios.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, two major parts of calculus. Essentially, it tells us that differentiation and integration are inverse processes. This theorem has two main parts:
1. The first part states that if a function is continuous over an interval \([a, b]\), then the integral of the function’s derivative over that interval equals the difference in the values of the function.

• Formally, if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then: \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]

2. The second part states that if we have a function that is the integral of some other function, then the derivative of this integral function is simply the original function.
These concepts are extremely powerful, allowing us to calculate definite integrals, like in the original exercise, by finding antiderivatives. The theorem is essential for understanding the relationship between a function and its derivatives, serving as the foundational underpinning of calculus analysis.