Question

In Exercises 13-18, (a) use Simpson's Rule with n = 4 to approximate the value of the integral and (b) find the exact value of the integral to check your answer. (Note that these are the same integrals as Exercises 1-6, so you can also compare it with the Trapezoidal Rule approximation.) $$\int_{0}^{2} x^{2} d x$$

Short answer

The Simpson's Rule approximation and the exact value of the integral are calculated. After that, the two values are compared to each other to see how close the approximation is to the exact value.

Step-by-step solution

Compute Simpson's Rule approximation

Simpson's Rule states that an approximate value of the definite integral \( \int_{a}^{b} f(x) \, dx \) can be obtained using the formula \( \frac{b - a}{3n} [f(a) + 4 \sum_{i=1}^{n/2} f(a+2i-1h) + 2 \sum_{i=1}^{n/2-1} f(a+2ih) + f(b)] \), where \( h = \frac{b - a}{n} \). Here, \( f(x) \) is the function to be integrated, \( [a, b] \) is the interval over which to integrate, and \( n \) is the number of subintervals to use (it must be even). For the integral \( \int_{0}^{2} x^{2} \, dx \), \( a = 0 \), \( b = 2 \), \( n = 4 \), \( f(x) = x^{2} \), and \( h = \frac{2 - 0}{4} = 0.5 \). Substituting these values into the formula and simplifying the expression gives the Simpson's approximation.

Calculate the exact value of the integral

The exact value of the integral \( \int_{0}^{2} x^{2} \, dx \) is found by finding the antiderivative (or indefinite integral) of \( x^{2} \), which is \( \frac{1}{3}x^{3} \). Then, apply the Fundamental Theorem of Calculus by subtracting the antiderivative evaluated at \( 0 \) from the antiderivative evaluated at \( 2 \). This gives the exact value of the integral.

Compare the approximate and exact values

By comparing the approximate value obtained using Simpson's Rule and the exact value found analytically, it will be possible to see how closely Simpson's Rule approximates the integral in this particular case. It will also allow checking the accuracy of your calculations.

Key concepts

Definite Integral

A definite integral is a fundamental concept in calculus that represents the accumulation of quantities, such as areas under a curve, and is defined for a continuous function over a closed and bounded interval. Mathematically, for a function f(x) continuous on the interval \[a, b\], the definite integral is represented as \(\int_{a}^{b} f(x) \, dx\).
When you calculate the definite integral, you're finding the total sum of f(x) with respect to x across the interval from a to b. In physical terms, if you were to plot f(x) on a graph, the definite integral would represent the area between the function's curve, the x-axis, and the vertical lines at x=a and x=b.
For example, the exercise asks to find the definite integral of x^2 from 0 to 2, which geometrically represents finding the area under the curve of x^2 from x=0 to x=2.
To improve comprehension, visualize the function x^2 as a parabola opening upwards and imagine the area under this curve from x=0 to x=2. This visual can be especially helpful for students who learn better with graphical representations.

Numerical Integration

Numerical integration involves approximate calculation methods for evaluating definite integrals, especially when an exact formula for the integral is difficult to obtain or when dealing with complex functions. Simpson's Rule, as applied in the given exercise, is one such technique.
Simpson's Rule is valuable because it allows us to approximate the area under the curve with high accuracy by using parabolic segments instead of straight-line segments, which the Trapezoidal Rule uses. It is particularly effective for functions that are smooth and whose true curvature can be closely matched by parabolic arcs.

Understanding Simpson's Rule

Simpson's Rule relies on dividing the area under the curve into an even number of equal-width slices and estimating the area in each slice with a parabola that passes through the top points of the slice. The value n, which must be even, denotes the number of subintervals we split the range into. The more subintervals used, the closer the approximation is to the exact integral.
With this method, students can approximate integrals that are otherwise complex to calculate, equipping them with a powerful tool for mathematical analysis in both theoretical and applied sciences.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) bridges the gap between differentiation and integration, two primary concepts in calculus. In essence, this theorem tells us that if we have a continuous function f(x) over an interval [a, b], then once we have found an antiderivative of f(x), we can calculate definite integrals.
FTC is divided into two parts:

• Part 1 establishes the relationship between integration and differentiation, indicating that the antiderivative of a function can be used to evaluate definite integrals.
• Part 2 allows one to compute the value of a definite integral by evaluating the difference of the antiderivative at the endpoints of the interval.

Applied to our example, once we determine the antiderivative of x^2, which is \(\frac{1}{3}x^3\), we use FTC Part 2 by inserting the boundary values (0 and 2) and subtracting to find the exact value of the definite integral.
Understanding FTC is crucial because it gives us a direct method to calculate the exact area under the curve for functions for which we can find the antiderivative, bypassing the need for numerical approximations like Simpson's Rule unless the function's antiderivative is not readily available or the integral is too complex to handle analytically.