Simpsons Rule and Trapezoidal Rule

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How do i find f(x)? and then how do i answer the question? thanks

You plug the values into the formula for calculating Simpson's rule and there you are.

First you need to look in your text book to discover what Simpson's rule is.

I can't remember it off hand, but it concerns taking two subintervals together. Here you'd use 1 -> 1.5 -> 2 as one pair of subintervals and 2 -> 2.5 -> 3 as the other. These are the x values.

Then the y values are the values of the function at each of these values, which have been provided for you.

It's now just a matter of arithmetical calculation.

Well...
U have 5 values...
so h= (3-1)/5=0.4

Use this , u can apply the formula of either trapezoidal or Simpsons to get the value of the whole integration..

Let y(i) be the value of F(x(i)) .

Then u just use the formula...

h/3(y0 + 4(y1+y3+y5+....) +2 (y2+y4+y6.....+ y(n-2))+ yn) ...Simpsons rule...

substitute the values....

And if u want to find F(x) ..u can use Newton Interpolation or Gauss Interpolation...

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Originally Posted by Misa-Campo

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How do i find f(x)? and then how do i answer the question? thanks

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The trapezoidal rule was derived fro the area of a trapezoid whose two bases, b1 and b2, are parallel, and are separated by a perpendicular distace or altitude, h:
A = (b1 +b2)/2]*h

In your given table, the bases are the y-values, and the h is 0.5, which is the constant interval in the x-values.
So in the 1st trapezoid, b1 = 5 and b2 = 1.
Hence A1 = (5+1)/2]*0.5 = 1.5

In the 2nd trapezoid, b1 = 1 and b2 = -2
so, A2 = (1 +(-2))/2]*0.5 = -0.25

In the 3rd trapezoid, b1 = -2, and b2 = 3, so,
A3 = (-2 +3)/2]*0.5 = 0.25

A4 = [(3 +7)/2]*0.5 = 2.5

Therefore, total area A = A1 +A2 +A3 +A4 = 1.5 -0.25 +0.25 +2.5 = 4 sq.units.

Using Simpson's Trapezoidal Rule,
A = (d/2)[h1 +2h2 +2h3 +2h4 +h5]
A = (0.5 /2)[5 +2(1) +2(-2) +2(3) +7]
A = 0.25[16]
A = 4 sq.units ---------------the same as above.

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Originally Posted by ticbol

The trapezoidal rule was derived fro the area of a trapezoid whose two bases, b1 and b2, are parallel, and are separated by a perpendicular distace or altitude, h:
A = (b1 +b2)/2]*h

In your given table, the bases are the y-values, and the h is 0.5, which is the constant interval in the x-values.
So in the 1st trapezoid, b1 = 5 and b2 = 1.
Hence A1 = (5+1)/2]*0.5 = 1.5

In the 2nd trapezoid, b1 = 1 and b2 = -2
so, A2 = (1 +(-2))/2]*0.5 = -0.25

In the 3rd trapezoid, b1 = -2, and b2 = 3, so,
A3 = (-2 +3)/2]*0.5 = 0.25

A4 = [(3 +7)/2]*0.5 = 2.5

Therefore, total area A = A1 +A2 +A3 +A4 = 1.5 -0.25 +0.25 +2.5 = 4 sq.units.

Using Simpson's Trapezoidal Rule,
A = (d/2)[h1 +2h2 +2h3 +2h4 +h5]
A = (0.5 /2)[5 +2(1) +2(-2) +2(3) +7]
A = 0.25[16]
A = 4 sq.units ---------------the same as above.

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I have a question if we are using Simpson's rule of 5 function values, how come we use substitute y into the formula? even when i substitute x values into the formula answer is still the same.

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Originally Posted by Misa-Campo

I have a question if we are using Simpson's rule of 5 function values, how come we use substitute y into the formula? even when i substitute x values into the formula answer is still the same.

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The old formula I know about Simpson's Rule is
A = (d/2)[h1 +hn +2(h2 +h3 +h4 =...+h(n-1)]
where hn is the last h
and h(n-1) is the second to the last h.

In your table, the d is in terms of x, and the h is in terms of y.

Or, again, y represents the parallel bases of the trapezoids. In the old formula, the h represents the parallel bases of the trapezoids.

Meaning, do not be limited to the specific variables used in any formula. Do understand what the formula means, so that you can use the formula using any variable. Do not memorize the exact variables; do memorize the meaning of the formula.