# Can Simpson's Rule be Applied to $$I_{35}$$?

• MHB
• karush
In summary, Simpson's Rule is used to find the exact answer for a given function when $n=4, 8, 16, 32$.
karush
Gold Member
MHB
do be done by Simpsons Rule for $n=4, 8, 16, 32$

$$I_{35}=\int_{0}^{4} \left(3{x}^{5}-8{x}^{3}\right)\,dx$$

before I even start on this and seeing the graph how can SR even be done on this?

First, let's get the exact answer so we know what to expect...W|A gives:

$$\displaystyle \int_0^4 3x^5-8x^3\,dx=1536$$

Okay, next, we need:

[box=green]
Simpson's Rule

$$\displaystyle \int_a^b f(x)\,dx\approx S_n$$

where

$$\displaystyle S_n=\frac{b-a}{3n}\left(f\left(x_0 \right)+4f\left(x_1 \right)+2f\left(x_2 \right)+\cdots+2f\left(x_{n-2} \right)+4f\left(x_{n-1} \right)+f\left(x_n \right) \right)$$

$$\displaystyle x_i=a+i\frac{b-a}{n}$$[/box]

We are given $a=0$ and $b=4$...and so let's begin with the first case $S_4$.

i) $n=4$.

With this value for $n$, we find that $x_i=i$. And so:

$$\displaystyle S_4=\frac{1}{3}\left(f(0)+4f(1)+2f(2)+4f(3)+f(4)\right)=\frac{1}{3}(0+4(-5)+2(32)+4(512)+2560)=\frac{4652}{3}=1550.\overline{6}$$

Can you proceed for the other values of $n$?

$$\displaystyle I_{35 }= \int_0^4 3x^5-8x^3\,dx=1536=S_n$$
$$\displaystyle S_n=\frac{b-a}{3n}\left(f\left(x_0 \right)+4f\left(x_1 \right)+2f\left(x_2 \right)+\cdots+2f\left(x_{n-2} \right)+4f\left(x_{n-1} \right)+f\left(x_n \right) \right)$$$$\displaystyle x_i=a+i\frac{b-a}{n}$$

$\displaystyle a=0 \ \ b=4$

i) $\displaystyle n=4 \ \ x_i=i$

$\displaystyle S_4=\frac{1}{3}\left(f(0)+4f(1)+2f(2)+4f(3)+f(4)\right)=\frac{1}{3}(0+4(-5)+2(32)+4(512)+2560)=\frac{4652}{3}=1550.\overline{6}$
---------------------------------
ii) $\displaystyle n=8 \ \ x_i=0+i\frac{4-0}{8} =\frac{i}{2}$

$\displaystyle S_8=\frac{1}{6} \left[ f(0)+4f\left(\frac{1}{2}\right) +2f\left(\frac{2}{2}\right) +4f\left(\frac{3}{2}\right) +2f\left(\frac{4}{2}\right) +4f\left(\frac{5}{2}\right) +2f\left(\frac{6}{2}\right) +4f\left(\frac{7}{2}\right) +f\left(\frac{8}{2}\right) \right]\approx 1537$

Ill stop here since the $n=16$ and $n=32$ would be done the same way

Last edited:

## 1. Can Simpson's Rule be applied to I35?

Yes, Simpson's Rule can be applied to any integration problem, including I35.

## 2. What is Simpson's Rule?

Simpson's Rule is a numerical integration method used to approximate the value of a definite integral. It uses quadratic curves to estimate the area under a curve rather than straight line segments, resulting in a more accurate approximation.

## 3. What are the advantages of using Simpson's Rule?

Simpson's Rule offers a more accurate estimation of a definite integral compared to other numerical integration methods, such as the trapezoidal rule. It also requires fewer subintervals to achieve the same level of accuracy, making it more efficient.

## 4. Can Simpson's Rule be used for both single and double integrals?

Yes, Simpson's Rule can be used for both single and double integrals. However, for double integrals, it is necessary to use a modified version known as Simpson's 2/3 rule.

## 5. How do I use Simpson's Rule to approximate the value of a definite integral?

To use Simpson's Rule, you first divide the interval of integration into an even number of subintervals. Then, apply the formula:
I ≈ (h/3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)]
where h is the width of each subinterval and n is the number of subintervals. This will give you an approximation of the definite integral, with a smaller value of h resulting in a more accurate approximation.

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