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Chapter 6: Problem 45

Since Simpson's Rule is exact for cubics (thirddegree polynomials), if you use Simpson's Rule to approximate the integral of a cubic, what value of \(n\) should you use?

Short Answer

Expert verified
Use \( n = 2 \) to exactly integrate a cubic with Simpson's Rule.

Step by step solution

01

Understanding Simpson's Rule

Simpson's Rule is a method for numerical approximation of definite integrals. It is particularly effective for polynomial functions up to the third degree, as Simpson's Rule can exactly integrate third-degree polynomials.
02

Identifying the Requirement for Exactness

Since Simpson's Rule is exact for integrals of cubic (third-degree) polynomials, we need to determine the appropriate number of panels (slices) needed for the integral's exactness.
03

Determining the Value of n

In Simpson's Rule, the value of \( n \), which represents the number of subintervals or slices, must be even. For a cubic function (polynomial of degree 3), Simpson's Rule can be exact using just two slices (or one whole segment divided into two with one interior fitting). Therefore, \( n = 2 \) to maintain the property of exactness for such functions.
04

Conclusion

For a cubic polynomial, using \( n = 2 \) in Simpson's Rule results in exact integration. Hence, an even number, specifically \( n = 2 \), should be used for exact results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Integration
Numerical integration is a technique used to calculate the integral of a function when an analytical solution is difficult or impossible to obtain. It's like breaking down a complex curve into simpler parts that we can easily measure.

Here are some highlights:
  • Numerical integration approximates the area under a curve, which is the definite integral of a function.
  • It's useful for complex functions or datasets where manual integration is tedious.
  • Common methods include the Trapezoidal Rule, Simpson's Rule, and others.
  • These methods slice the region into intervals and approximate the area using basic geometric shapes like rectangles or parabolas.
Numerical integration, thus, provides a pragmatic approach to solving real-world problems where precision is needed, but exact solutions are hard to find.
Third-degree Polynomials
A third-degree polynomial, also known as a cubic polynomial, is a polynomial of the form:\[ f(x) = ax^3 + bx^2 + cx + d \]where \( a eq 0 \). This polynomial has the following characteristics:
  • It can have up to three roots, or solutions, where the polynomial equals zero.
  • Its graph is a curve that can have one or two turning points, depending on the values of the coefficients.
  • The degree of the polynomial tells us how many real roots the function might have, maximizing out at three.
Third-degree polynomials are significant in numerical integration because Simpson's Rule can exactly integrate them. This makes the rule especially valuable for practical applications involving cubic equations.

Understanding cubic polynomials helps in grasping why certain integration techniques are effective, particularly when the degree of the polynomial matches the expected conditions for exactness, like in Simpson's Rule.
Approximate Integrals
Approximate integrals involve computing the integral of a function by estimating the area under its curve. This is often done when finding the exact integral is not feasible. Here鈥檚 what you should know:
  • Approximation gives us a close estimate of the integral, improving efficiency in complex calculations.
  • Methods like Simpson鈥檚 Rule are used because they strike a balance between simplicity and accuracy.
  • These methods can be particularly accurate when dealing with lower degree polynomials, such as cubics.
  • Simpson鈥檚 Rule, for instance, is famous for its precision with third-degree polynomials, offering exact results with just a few intervals.
These methods test the clever balancing of computational efficiency and precision in practice. Even though they are termed 'approximations,' they can yield results that are exact under specific conditions, like with cubic functions and suitable values of \( n \), such as 2 in Simpson's Rule. Utilizing approximation effectively is key in fields requiring precise measurement and calculation.

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Most popular questions from this chapter

A telephone company estimates the maximum market for car phones in a city to be 10,000 . Total sales are proportional to both the number already sold and the size of the remaining market. If 100 phones have been sold at time \(t=0\) and after 6 months 2000 have been sold, find a formula for the total sales after \(t\) months. Use your answer to estimate the total sales at the end of the first year.

A hydroelectric dam generates electricity by forcing water through turbines. Sediment accumulating behind the dam, however, will reduce the flow and eventually require dredging. Let \(y(t)\) be the amount of sediment (in thousands of tons) accumulated in \(t\) years. If sediment flows in from the river at the constant rate of 20 thousand tons annually, but each year \(10 \%\) of the accumulated sediment passes through the turbines, then the amount of sediment remaining satisfies the differential equation \(y^{\prime}=20-0.1 y\). a. By factoring the right-hand side, write this differential equation in the form \(y^{\prime}=a(M-y)\). Note the value of \(M\), the maximum amount of sediment that will accumulate. b. Solve this (factored) differential equation together with the initial condition \(y(0)=0\) (no sediment until the dam was built). c. Use your solution to find when the accumulated sediment will reach \(95 \%\) of the value of \(M\) found in step (a). This is when dredging is required.

True or False: If \(\int_{-\infty}^{x} f(x) d x\) converges, then \(\int_{0}^{x} f(x) d x\) converges.

Use a graphing calculator to estimate the improper integrals \(\int_{0}^{x} \frac{1}{x^{2}+1} d x\) and \(\int_{0}^{x} \frac{1}{\sqrt{x}+1} d x\) (if they converge) as follows: a. Define \(y_{1}\) to be the definite integral (using FnInt) of \(\frac{1}{x^{2}+1}\) from 0 to \(x\). b. Define \(y_{2}\) to be the definite integral of \(\frac{1}{\sqrt{x}+1}\) from 0 to \(x\). c. \(1 / 1\) and \(y_{2}\) then give the areas under these curves out to any number \(x\). Make a TABLE of values of \(y_{1}\) and \(y_{2}\) for \(x\) -values such as 1,10 , 100,500, and 10,000 . Which integral converges (and to what number, approximated to five decimal places) and which diverges?

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. \(y^{\prime}=0.4 y(0.01-y)\)
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