Problem 2 - Essential Calculus

Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{1}^{2} \frac{2}{x^{2}} d x, \quad n=4 $$

Step-by-Step Solution

Answer

The definite integral approximated using the Trapezoidal Rule and Simpson's Rule, with \(n = 4\), should yield similar values for the exercise. The calculations involved in the process will provide the specific values

Step 1: Express the function in terms of x

Firstly, convert the integral into a function in terms of x. So the function \(f(x) = \frac{2}{x^{2}} = 2x^{-2}\)

Step 2: Calculate the width of the subinterval

Next, calculate the width of each subinterval using the formula \(h = \frac{b - a}{n}\). With \(a = 1\), \(b = 2\), and \(n = 4\), we have \(h = \frac{2 - 1}{4} = 0.25\) meaning each subinterval is 0.25 units wide.

Step 3: Apply the Trapezoidal Rule

The Trapezoidal Rule is given by \(T_{n} = \frac{h}{2} [f(a) + 2(f(a+h) + f(a+2h) + f(a+3h)) + f(b)]\). Substitute the respective values into the equation and simplify. \(T_{4} = \frac{0.25}{2} [f(1) + 2(f(1.25) + f(1.5) + f(1.75)) + f(2)]\). This will give an approximation for the integral.

Step 4: Apply Simpson's Rule

Simpson's Rule is given by \(S_{n} = \frac{h}{3} [f(a) + 4(f(a+h) + f(a+3h)) + 2(f(a+2h)) + f(b)]\). Again substitute the respective values into the equation and simplify. \(S_{4} = \frac{0.25}{3} [f(1) + 4(f(1.25) + f(1.75)) + 2(f(1.5)) + f(2)]\). Calculate this value to get an approximation for the integral.

Step 5: Compare the results

Compare the results from the Trapezoidal Rule and Simpson's Rule. These values should be close to the actual value of the definite integral.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a method of numerical integration used to approximate the definite integral of a function. This rule transforms the area under a curve into a series of trapezoids and then calculates the area of each trapezoid to find an approximation of the entire integral. Visualize the curve between two points on the x-axis, and draw straight lines to connect the points on the curve to form trapezoids. Mathematically, the formula for the Trapezoidal Rule with n subintervals is given by:
\[T_n = \frac{h}{2} [f(a) + 2(f(a+h) + f(a+2h) + \ldots + f(a+(n-1)h)) + f(b)]\]
where \( h \) is the width of each subinterval, and \( f(a + kh) \) represents the function's values at different subinterval points. Though this method is simple and quite intuitive, its accuracy depends on the number of subintervals used; more subintervals usually result in a better approximation.

Simpson's Rule

Simpson's Rule provides an alternative to the Trapezoidal Rule, offering a more accurate approximation of a definite integral for functions that are reasonably smooth. This rule fits a parabola to each pair of subintervals and finds the area under each parabola to approximate the integral's value. Each set of three points along the curve contributes to a parabolic arc between them. The formula for Simpson's Rule when n is even is as follows:
\[S_n = \frac{h}{3} [f(a) + 4(f(a+h) + f(a+3h) + \ldots) + 2(f(a+2h) + f(a+4h) + \ldots) + f(b)]\]
In this equation, the terms multiplied by 4 are the odd indices representing the midpoints of the subintervals, while the multiplication by 2 represents the even indices except for the first and last point. Students should remember that for Simpson's Rule to be applicable, the number of subintervals (n) must be even.

Numerical Integration

Numerical integration comprises various algorithms for approximating the definite integral when an exact formula is hard to come by or when dealing with empirical data. Aside from the Trapezoidal and Simpson's Rules, there are other methods like the Midpoint Rule, and Monte Carlo integration, each with its own advantages and contexts for use. In practice, numerical integration allows for the analysis of complex and real-world problems in physics, engineering, and other fields where it is impossible to calculate the integral analytically with basic calculus techniques. While more accurate methods exist for numerical integration, they may require complex computations or software, making the Trapezoidal and Simpson’s Rules popular for their balance of simplicity and accuracy.

Calculus

Calculus is a branch of mathematics focused on change and motion. Its two main types are differential calculus, concerning rates of change and slopes of curves, and integral calculus, involving accumulation of quantities and the areas under and between curves. Definite integrals, the focus of both the Trapezoidal and Simpson's Rule, are a central concept in integral calculus often representing physical quantities such as distances, areas, and volumes. The ability to approximate integrals using methods like the Trapezoidal and Simpson’s Rule not only reinforces the concepts of calculus but also serves as a bridge between theoretical mathematics and practical applications in various scientific and engineering disciplines.