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Chapter 6: Problem 46

What are the "weights" for Simpson's Rule when \(n=2 ?\)

Short Answer

Expert verified
The weights for Simpson's Rule when \(n=2\) are 1, 4, 1.

Step by step solution

01

Understand the Question

Simpson's Rule is a method for numerical integration. When applying Simpson's Rule, weights are associated with the function values to approximate the integral. For \(n = 2\), it refers to using two subintervals, which means three points: starting point, midpoint, and endpoint.
02

Formula for Simpson's Rule

Simpson's Rule formula can be stated as \( \int_a^b f(x) \, dx \approx \frac{b-a}{6} [f(a) + 4f(m) + f(b)] \). Here, \(a\) and \(b\) are the endpoints of the interval, and \(m\) is the midpoint. The coefficients 1, 4, and 1 are the weights.
03

Identify the Weights

For \(n = 2\), referring to the notation \(f(a)\), \(f(m)\), and \(f(b)\), the weights in Simpson's Rule are \(1\), \(4\), and \(1\). This ensures that the quadratic approximation used by Simpson's Rule is appropriately applied.

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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 fundamental concept in mathematics used to find approximate solutions to definite integrals. When the exact integral of a function is difficult or impossible to compute analytically, numerical integration comes into play. It's a powerful tool that helps us estimate the area under a curve or solve complex integral equations.
  • Numerical methods break down the integration process into simpler steps, using basic calculations.
  • Simpson's Rule, for example, is one such method that simplifies the integration of polynomial functions.
  • These techniques often rely on a chosen number of subintervals or points, as accuracy can vary based on this selection.
Methods such as Simpson's Rule are not just arbitrary; they are grounded in mathematical calculations to provide the best possible approximation given the constraints.
Weights in Integration
In numerical integration, weights play a crucial role. They are essentially the coefficients used in integration formulas to scale the contributions of function evaluations at specific points. Understanding weights is key to grasping how different numerical methods like Simpson's Rule work.
  • Weights determine the influence each function value has in approximating the integral.
  • For Simpson's Rule with three points, the weights are 1, 4, and 1. These weights emphasize the midpoint more heavily, aligning with the method's mathematical justification.
  • This influence helps achieve a more accurate representation of the integral, particularly for parabolic approximations.
Without proper weighting, the integration might either underestimate or overestimate the true value, especially when dealing with non-linear functions.
Quadratic Approximation
Quadratic approximation is a mathematical technique used to approximate functions using parabolas. It's closely related to Simpson's Rule, where a parabolic curve is used to approximate the section of the graph between selected points.
  • Simpson's Rule approximates the curve by fitting parabolas to smaller intervals of the function.
  • This approach is valuable because quadratic functions (parabolas) can more accurately represent many real-world phenomena than linear functions.
  • The method's accuracy is significantly high for smooth functions because the curvature of a parabola can closely match the actual curvature of the function being integrated.
By using quadratic approximation in numerical integration, Simpson's Rule leverages these mathematical characteristics to enhance precision, making it a preferred choice in many practical applications.

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Most popular questions from this chapter

Suppose that you now have $$\$ 6000$$, you expect to save an additional $$\$ 3000$$ during each year, and all of this is deposited in a bank paying \(10 \%\) interest compounded continuously. Let \(y(t)\) be your bank balance (in thousands of dollars) \(t\) years from now. a. Write a differential equation that expresses the fact that your balance will grow by 3 (thousand dollars) and also by \(10 \%\) of itself. [Hint: See Example 7.] b. Write an initial condition to say that at time zero the balance is 6 (thousand dollars). c. Solve your differential equation and initial condition. d. Use your solution to find your bank balance \(t=25\) years from now.

Think of the slope field for the differential equation \(\frac{d y}{d x}=\frac{6 x}{y^{2}} .\) What is the sign of the slope in quadrant I (where \(x\) and \(y\) are both positive)? What is the sign of the slope in each of the other three quadrants? Check your answers by looking at the slope field on page 463 .

In the reservoir model, the heart is viewed as a balloon that swells as it fills with blood (during a period called the systole), and then at time \(t_{0}\) it shuts a valve and contracts to force the blood out (the diastole). Let \(p(t)\) represent the pressure in the heart at time \(t\) a. During the diastole, which lasts from \(t_{0}\) to time \(T, p(t)\) satisfies the differential equation $$\frac{d p}{d t}=-\frac{K}{R} p$$ Find the general solution \(p(t)\) of this differential equation. ( \(K\) and \(R\) are positive constants determined, respectively, by the strength of the heart and the resistance of the arteries. The differential equation states that as the heart contracts, the pressure decreases \((d p / d t\) is negative) in proportion to itself.) b. Find the particular solution that satisfies the condition \(p\left(t_{0}\right)=p_{0} . \quad\left(p_{0}\right.\) is a constant representing the pressure at the transition time \(t_{0}\).) c. During the systole, which lasts from time 0 to time \(t_{0}\), the pressure \(p(t)\) satisfies the differential equation $$\frac{d p}{d t}=K I_{0}-\frac{K}{R} p$$ Find the general solution of this differential equation. \(\left(I_{0}\right.\) is a positive constant representing the constant rate of blood flow into the heart while it is expanding.) [Hint: Use the same \(u\) -substitution technique that was used in Example 7.] d. Find the particular solution that satisfies the condition \(p\left(t_{0}\right)=p_{0}\). e. In parts (b) and (d) you found the formulas for the pressure \(p(t)\) during the diastole \(\left(t_{0} \leq t \leq T\right)\) and the systole \(\left(0 \leq t \leq t_{0}\right)\). Since the heart behaves in a cyclic fashion, these functions must satisfy \(p(T)=p(0)\). Equate the solutions at these times (use the correct formula for each time) to derive the important relationship $$ R=\frac{p_{0}}{I_{0}} \frac{1-e^{-K T / R}}{1-e^{-K t_{0} / R}} $$

$$ \text { Use integration by parts to find each integral. } $$ $$ \int s(2 s+1)^{4} d s $$

$$ \text { Use integration by parts to find each integral. } $$ $$ \int t e^{-0.2 t} d t $$
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