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Questions tagged [simpsons-rule]

For questions regarding Simpson's rule for approximating the integral of a function using quadratic polynomials, and its applications.

Simpson's rule is a technique for estimating integrals numerically, and can be written as

$$\int_a^b f(x) dx \approx \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]$$

Simpson's rule can be derived by using a quadratic approximation for $f$, or it can be viewed as a weighted average of the trapezoid and midpoint rules. The error involved can be bounded above by

$$\frac{1}{90} \left(\frac{b - a}{2}\right)^5 \left|f^{(4)}(\xi)\right|$$

where $\xi$ is a number between $a$ and $b$.

Reference: Simpson's rule.

142 questions
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Error bound for $\cos(x^3)$ under Mn, Tn and Sn

In this problem we will approximate the integral of $\cos(x^3)$ over the interval $[0, 2]$. (a) Write an expression for MN, TN and SN with $N = 4$. (b) For each of the approximations determine an N so that the error is guaranteed to be less than…
user77117
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How to apply Simpson's rule to f(x)^2*x?

I have points available for $x$ and $f(x)$. To calculate area between $(f(x)^2)\cdot(x)$ and $x$, can I consider $g(x) = (f(x)^2)\cdot(x)$ and apply Simpson's rule on $g(x)$ and $x$?
Peeyush Gupta
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Simpson's rule using Taylor polynomial

From a proof of Simpson's rule using Taylor polynomial where $f\in[x_{0},x_{2}]$ and, for $$x_{1}=x_{0}+h$$ where $$h=\frac{x_{2}-x_{0}}{2}$$, it…
mvfs314
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Doubling Number of Nodes In Composite Simpson's Rule

Let n be even. Show how the composite Simpson rule with 2n equally spaced nodes can be computed from the case of n equally spaced nodes with a minimum amount of additional work. I've been working on this problem for almost 2 hours now and can't seem…
James Snyder
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How can we tell if Simpson’s rule is an underestimate or overestimate of the area under a curve?

Context: high school math Question: integrating a semicircle $y= \sqrt{1-x^2}$ from $0$ to $1$ using five function values with Simpson’s rule gives me $0.7709$. The exact value is $0.785$. I understand for the trapezoidal rule we can predict if it…
Nimna De Silva
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How to use Simpsons Rule Maths for Ship Stability

I did a Civil Engineering course some years ago and from my textbook I had this question. As I am interested in this I have been trying to solve this, but unfortunately I haven't been able to find a good solution yet. I would be appreciative if…
Rob Wilkinson
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Simpsons rule problem. why is this equation setup this way?

I am a bit confused as to why a problem in my book is using A(t) instead of D(t) in teh setup for simpsons rule. Why is the integral at the end setup like: $$\int_0^43200 A(t)dt$$ and not $$\int_0^43200 D(t)dt$$ Here are the passages:
Jwan622
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Estimating with simpson rule

I have a question that is supposed to be very easy in a test: We approximate $\displaystyle \int_0^1 x^2$ with Simpsonrule, and 5 intervals. Choose solution: Firstly I don't understand what is meant with 5 intervals. Are the intervals not supposed…
Dovendyr
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Give a function for which Simpson's rule returns an exact value

Let I denote the integral $I = \int_0^{\pi/2}\sqrt{\sin x}dx$ and 4 strips. Give a function for which Simpson’s rule returns an exact value. I just entered in the exact values (so $\sin(\pi/2)$ for example) into the formula but did not get an exact…
user639649
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Step in proof: error Simpson's rule for $f\in C^4$

They claim later on in the proof that $G’(0)=0$. I don’t see this. I would say that we have $$ G(t)=\int_0^tF(\tau)d\tau-\int_0^{-t}F(\tau)d\tau-t/3[F(-t)+4F(0)+F(t)], $$ which yields $$ G’(0)=F(0)-F(0)-1/3[6\cdot F(0)]=-2F(0). $$ Could someone…
Sha Vuklia
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Trapezoidal and simpson rule question here?

The trapezoidal rule applied on $ \int_0^{2} [f(x)] dx$ gives the value 5 and the Midpoint rule gives the value 4. What value does Simpson's rule give? So we have that T=f(0)+f(2). The Simpson's value is S=(1/3)*(f(0) + 4f(1)+ f(2)) f(1) here is…
user3543012
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