Question

what are the meaning of Trapezoidal rule and simpson's rule??

Answer 1

The area under a curve can be approximated with trapezoids and arcs. They are geometric approaches to approximating integrals

Answer 2

yes..i know that trapezoidal rule and simson's rule is to compute the approximate value and it is the technique of approximating definite integral... How can you relate or define the truncation error and the round off error to the two rule?

Answer 3

The approx is less than the actual area, im not sure what you would need as far as a rigorous answer.

Answer 4

I'm sorry i did not explain what ineed to you ..i need some ideas for this topic..i hope you help me?? can you please give me the difference between a truncation error and a round off error when appliying to either the trapezoidal rule or simpson's rule??

Answer 5

ill have to research it

Answer 6

wiki implies that the two terms are interchangeable
"Occasionally, round-off error (the consequence of using finite precision floating point numbers on computers) is also called truncation error, especially if the number is rounded by truncation." http://en.wikipedia.org/wiki/Truncation_error

Answer 7

When ever ive computed Simpsons rule values (in calc two) we arbitrarily defined N to the be the number of sections the curve was divided into, the error would be in this N being finite as opposed to infinite.

Answer 8

If your class has a formal definition for these concepts perhaps you could use integrals that are easy to compute in order to better understand the accuracy of these approximation methods

Answer 9

Can you answer this question?
State the trapezoidal rule for approximating the value of definite integral of a function "f" on the closed interval [a,b], what necessary condition must function "f" satisfying in order to applying the trapezoidal rule interpret the trapezoidal rule in terms of geometry??

Answer 10

Yes, my classmate discuss this method, but some details i cannot understand. they give the formula and how to derived,,then she give some example to apply this method..sometimes, i can solve the equation i cannot get the exact value? and i don't know why? can you give me some example that i can solve it and can you please check it if is correct or wrong??

Answer 11

imagine a function that is a straight line, the trapezoid here is a rectangle and the area is length times width right? so as the curve differs from a straight line, more and more trapezoids are used, but it is the same idea as being able to approximate as though it was a rectangle... http://en.wikipedia.org/wiki/Trapezoidal_rule

Answer 12

I can see the website that you give.,,i read it but i want example from to you then i can't solve it..in my own and i hope you will help me to understand it clearly.

Answer 13

Okey i can solve it now

Delta X = (b-a)/n = (1-0)/5 = 1/5
the value of Xo to X5:
Xo=0, X1=1/5, X2=2/5, X3= 3/5, X4= 4/5, X5=1

\[\int\limits_{0}^{1}f(x)dx=(b-a)/2n[f(xo)+2f(x1)+2f(x2)+2f(x3)....+f(Xn-1)+f(Xn)]=(1-0)/10[0+50+12.5+5.56+3.125+2)=1/10(73.185)=7.3185\]
In integration the: \[\int\limits_{0}^{1}1/X^2dx = 1\]

find the error: get the second to third derivative and substitute the value of x or the closed interval (0,1) to the second derivative...then substitute to the formula of the truncation error.
\[f(x) =1/X^2, f'(x)=1/2x^3, f''(x)= 6/x^4,f'''(x)=1/24x^5\]\[f''(0) = 6/0^4=0 ; f''(1)=6/1^4=6\]

now solve the error:
\[\epsilon _{T} = -1/12(b-a)f''(\eta)\Delta _{x}=-1/12(1-0)f''(0)(1/5)=-1/12(1)(0)(1/5)=0
\[\[\epsilon _{T} = -1/12(b-a)f''(\eta)\Delta _{x}=-1/12(1-0)f''(1)(1/5)=-1/12(1)(1)(1/5)^2=-1/300=-0.00333\]\]

Answer 14

\[\int\limits_{0}^{1}1/x^2dx ; n=5\]

Answer 15

First get the delta x...then the value of Xo to X5..because of n=5..then solve it using the formula of trapezoidal rule, then u get the approximation of that equation. then to get the error used the formula of truncation error.