Question

Prove that Simpson's Rule is exact when used to approximate the integral of a cubic polynomial function, and demonstrate the result for \(\int_{0}^{1} x^{3} d x, n=2\).

Short answer

Simpson's Rule provides an exact solution for the integral of a cubic polynomial function. This was demonstrated with \(\int_{0}^{1} x^{3} dx\), for which the Simpson’s Rule also yields the exact value 0.25.

Step-by-step solution

Derivation of Simpson's Rule

Simpson’s Rule states that \[ \int_{a}^{b}f(x)dx = \frac{(b-a)}{6}[f(a)+4f((a+b)/2)+f(b)]\n\]Simpson's Rule is derived by integrating the quadratic polynomial that passes through the points (a, f(a)), ((a + b) / 2, f((a+b)/2)), and (b, f(b)).

Application of Simpson's Rule to a cubic polynomial

If f(x) is a cubic polynomial, i.e., f(x)=ax^3 + bx^2 + cx + d, then Simpson's Rule gives the exact integral because integrating a cubic polynomial from a to b does not add higher than cubic terms, and the rule is already exact for any quadratic polynomial.

Proving Exactness

The Simpson’s Rule is given by the formula\[ \int_{x0}^{x2}f(x)dx = \frac{h}{3}[f(x0)+4f(x1)+f(x2)]\n\]where \(h = (x2 - x0)/2\), and \(x1\) is the midpoint between \(x0\) and \(x2\). If \(f(x)\) is a cubic polynomial, then we have \[f(x) = ax^3 + bx^2 + cx + d\]Plugging this into the Simpson’s Rule formula, the cubic terms cancel out and we are left only with lower order terms, which proves that Simpson’s Rule is exact for cubic polynomials.

Demonstrate the Result

The cubic function is \(f(x) = x^3\) and its integral from 0 to 1 is \(\int_{0}^{1} x^{3}dx\), which is 0.25. With n=2, implying a=0, m=0.5, b=1, one application of Simpson's Rule yields\[\int_{0}^{1} x^{3} dx = \frac{(1-0)}{6}(f(0)+4f(0.5)+f(1)) = \frac{1}{6}(0 + 4 * (0.5)^3 + 1) = 0.25\] The answer matches the exact value of the integral, which confirms that Simpson's Rule is indeed exact for the integral of a cubic function.

Key concepts

Cubic Polynomial

A cubic polynomial is a function of the form \(f(x) = ax^3 + bx^2 + cx + d\). Here, \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\). Cubic polynomials are important because they are the simplest type of polynomial that can form a curve with a single bend or point of inflection. This characteristic makes them interesting to study because they can model more complex behaviors than linear or quadratic polynomials.

In our example, we're dealing specifically with the cubic polynomial \(f(x) = x^3\). When integrating or approximating integrals using methods like Simpson's Rule, it’s crucial to understand the behavior of such polynomial functions. They play a pivotal role in more advanced mathematical concepts and numerical methods.

Integration

Integration is a fundamental concept in calculus which involves finding the integral of a function. This integral represents the area under the curve of the function in a given interval. Integration is particularly essential when it comes to calculating areas, volumes, and other quantities under complex curves.

For a cubic polynomial like \(x^3\), the process of integration allows us to determine the accumulated quantity from one point to another, effectively finding the net area under the curve. In our exercise example, the exact integral of \(x^3\) from 0 to 1 is 0.25. This result signifies the efficient power of calculus in determining precise values when direct measurement is unfeasible.

Understanding integration deepens our grasp of how calculus is applied in real-world contexts and in developing numerical methods.

Exactness of Numerical Methods

The exactness of numerical methods refers to how accurately a numerical approximation can measure or represent an integral. In terms of numerical integration techniques like Simpson's Rule, exactness is achieved when the estimated value equals the actual integral value, with no error in the measurement.

For a cubic polynomial, Simpson’s Rule achieves this rare degree of precision because the rule is tailored to match the behavior of polynomials up to the third degree.

This accuracy hinges on the fact that Simpson’s Rule uses midpoints and endpoint values to approximate a more complex curve with complete fidelity for up to cubic terms. This inherent exactness makes Simpson's Rule particularly valuable for polynomial integrals up to the third degree, providing a potent tool in scenarios where exact calculations are needed rapidly without the rigors of manual integration.

Quadratic Polynomial

A quadratic polynomial refers to functions of the form \(f(x) = ax^2 + bx + c\). Quadratic functions are simpler than cubic polynomials and typically exhibit a parabolic curve, either opening upwards or downwards depending on the sign of \(a\).

Simpson's Rule is based on the concept of fitting such a quadratic polynomial through specific points of the curve and using the resultant area as the estimate for the integral. This reliance on quadratic polynomials allows Simpson's Rule to offer more accurate integrations than simpler methods like the trapezoidal rule, especially for functions that themselves approximate a parabola within small intervals.

By ensuring this basis, Simpson's Rule achieves its exactness for quadratic, and consequently, cubic polynomials, highlighting the powerful synergy between these polynomials and numerical integration techniques.

Numerical Integration

Numerical integration encompasses a range of methods for calculating the integral of functions when analytical integration is challenging or impractical. These methods break down complex curves into simpler shapes that are easier to measure.

In the context of Simpson’s Rule, numerical integration is performed by approximating the area under a curve through the use of parabolas. This approach involves partitioning the interval and using particular points to fill a scheme that best represents the overarching curve.

Methods like Simpson’s Rule are invaluable, especially when dealing with functions where computing the integral directly is cumbersome. They offer practical answers with a suitable degree of accuracy, crucial for fields that require quick but reliable computational solutions. Numerical integration, therefore, provides an essential bridge between theory and practical applications.