Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{0}^{8} \sqrt[3]{x} d x, n=8 $$

Step-by-Step Solution

Answer

The solutions from the Trapezoidal Rule and Simpson's Rule will be approximations of the exact integral, which can be found using the antiderivative of \( \sqrt[3]{x} \). The comparisons of these approximations with the exact value will be key.

Step 1: Applying the Trapezoidal Rule

For a function \( f(x) \) and an interval \([a, b]\), the Trapezoidal Rule can be written as \((b-a) \times \frac{f(a) + f(b)}{2}\). Here, \( a=0 \), \( b=8 \), and \( f(x)=\sqrt[3]{x} \). Therefore, we begin by dividing the interval into 8 equal parts to obtain a sequence of subintervals and evaluate the function at these points.

Step 2: Applying Simpson's Rule

Simpson's Rule can be written as: \(\frac{b - a}{6} [f(a) + 4f\left(\frac{a+b}{2}\right) + f(b)]\) for an interval \([a, b]\). Again, divide the interval into 8 equal parts as done in Step 1, and evaluate the function at these points. We sum the evaluations to estimate the integral.

Step 3: Calculate the Exact Integral

The exact integral can be evaluated using the fundamental theorem of calculus. In this case, the antiderivative of \( \sqrt[3]{x} \) is \(\frac{3}{4}x^{\frac{4}{3}}\). We find the exact integral by evaluating the antiderivative at the bounds 8 and 0, and subtracting these results.

Step 4: Comparison

Finally, we compare these approximations obtained using the Trapezoidal Rule, Simpson’s Rule, and the exact value of the integral.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule offers one way to approximate the value of a definite integral. This rule turns out to be particularly handy when we are dealing with functions that are difficult or impossible to integrate analytically. The general idea is to divide the area under the curve into a series of trapezoids rather than perfect rectangles, as is done in a Riemann sum approximation. The formula for applying the Trapezoidal Rule over one interval is \br \[ (b-a) \times \frac{f(a) + f(b)}{2} \], \br where \( a \) and \( b \) are the interval endpoints, and \( f(a) \) and \( f(b) \) are the function values at those endpoints. To use this rule for our exercise, we first divide the interval from 0 to 8 into 8 equal parts and then calculate the sum of the areas of the trapezoids formed. With more intervals, or a higher value of \( n \), we can achieve a more accurate approximation.

Simpson's Rule

Simpson's Rule is another method for numerical integration that often yields more accurate results than the Trapezoidal Rule for the same number of intervals. Simpson's Rule is derived by approximating the curve with parabolic arcs instead of straight lines. The basic formula over two intervals is \br \[ \frac{b - a}{6} \big(f(a) + 4f(\frac{a+b}{2}) + f(b)\big) \], \br In this context, we again split the interval from 0 to 8 into equal parts, but this time we apply the Simpson’s Rule which involves more function evaluations including the midpoint. For our given exercise, we use 8 intervals and the corresponding formula, including weighted function values, to estimate the integral. Much like the Trapezoidal Rule, increasing the number of intervals (or \( n \)) in Simpson's Rule will generally lead to a better approximation of the integral.

Fundamental Theorem of Calculus

Understanding the Fundamental Theorem of Calculus is pivotal to fully grasp the concept of integration. It bridges the concept of a derivative with that of an integral. Essentially, the Fundamental Theorem of Calculus states that if a function \( f \) is continuous over an interval \( [a, b] \) and \( F \) is an antiderivative of \( f \) over that interval, then \br \[ \int_{a}^{b} f(x) dx = F(b) - F(a) \]. \br By evaluating the antiderivative at the endpoints and subtracting, we can find the exact value of a definite integral. In our exercise, we used this theorem to calculate the exact integral of \( \sqrt[3]{x} \), resulting in the antiderivative \( \frac{3}{4}x^{\frac{4}{3}} \). We then evaluated this expression at 8 and 0, providing the exact area under the curve from 0 to 8.

Numerical Integration

When we talk about numerical integration, we're referring to a spectrum of algorithms and techniques used to approximate the value of definite integrals, especially when an exact solution is elusive. Numerical methods, including the Trapezoidal Rule and Simpson's Rule, excel when the integral is too complex for analytical solutions or when dealing with empirical data where we don't have a functional form. The goal of these methods is to provide a good approximation to the integral by using sums that involve values of the function at specific points. Various numerical integration methods offer trade-offs between computational complexity and accuracy. For example, while the Trapezoidal Rule is straightforward and easy to apply, Simpson's Rule typically gives a more precise estimate but requires more function evaluations. Given multiple methods, we can choose the most suitable one based on the function we're integrating and the required precision.