Composite form of Boole's Rule

[says]

Homework Statement

Develop a composite form of Boole's rule for an integral of the form ∫ f(x) dx, where the bounds of integration are from [a,b].

Determine the error bound formula for the composite form of Boole's rule.

∫ g(t) dt = h/45[14g(0)+64g(h)+24g(2h)+64g(3h)+14g(4h)] - (8h⁷/945)*d⁶g/dt⁶ (ξ)

for some ξ ∈ [0,4h]
bounds of integration are [0,4h]

Homework Equations

∫ g(t) dt [bounds of integration [a,b]
a=a
b=a+nh

The Attempt at a Solution

∫ g(t) dt = h/45[14g(0)+64g(h)+24g(2h)+64g(3h)+14g(4h)]

∫ g(t) dt = 2h/45[7g(0)+32g(h)+12g(2h)+32g(3h)+7g(4h)]

∫ g(t) dt = 2h/45[7g(a)+32g(a+h)+12g(a+2h)+32g(a+3h)+7g(a+4h)]

∫ g(t) dt = 2h/45[7g(a)+32g(a+h)+12g(a+2h)+32g(a+3h)+7g(a+4h)]

∫ g(t) dt = 2h/45[7g((a)+(a+4h))+32g((a+h)+(a+3h))+12g(a+2h)]

I think that is the composite of Boole's rule. I'm not sure how to determine the error bound formula for the composite form of Boole's rule though. Any help would be much appreciated :)

 

[Ray Vickson]

Homework Statement

Develop a composite form of Boole's rule for an integral of the form ∫ f(x) dx, where the bounds of integration are from [a,b].

Determine the error bound formula for the composite form of Boole's rule.

∫ g(t) dt = h/45[14g(0)+64g(h)+24g(2h)+64g(3h)+14g(4h)] - (8h⁷/945)*d⁶g/dt⁶ (ξ)

for some ξ ∈ [0,4h]
bounds of integration are [0,4h]

Homework Equations

∫ g(t) dt [bounds of integration [a,b]
a=a
b=a+nh

The Attempt at a Solution

∫ g(t) dt = h/45[14g(0)+64g(h)+24g(2h)+64g(3h)+14g(4h)]

∫ g(t) dt = 2h/45[7g(0)+32g(h)+12g(2h)+32g(3h)+7g(4h)]

∫ g(t) dt = 2h/45[7g(a)+32g(a+h)+12g(a+2h)+32g(a+3h)+7g(a+4h)]

∫ g(t) dt = 2h/45[7g(a)+32g(a+h)+12g(a+2h)+32g(a+3h)+7g(a+4h)]

∫ g(t) dt = 2h/45[7g((a)+(a+4h))+32g((a+h)+(a+3h))+12g(a+2h)]

I think that is the composite of Boole's rule. I'm not sure how to determine the error bound formula for the composite form of Boole's rule though. Any help would be much appreciated :)

I think that what the question wants you to find is a formula for $\int_a^b f(x) \, dx$ obtained by splitting the interval $[a,b]$ into $4n$ pieces of length $h = (b-a)/(4n)$ each. So, instead of having just 4 intervals you might have 40 or 400 intervals---some multiple of 4, anyway.

 

[says]

I think that what the question wants you to find is a formula for ∫baf(x)dx∫abf(x)dx\int_a^b f(x) \, dx obtained by splitting the interval [a,b][a,b][a,b] into 4n4n4n pieces of length h=(b−a)/(4n)h=(b−a)/(4n)h = (b-a)/(4n) each. So, instead of having just 4 intervals you might have 40 or 400 intervals---some multiple of 4, anyway.

Ok, so I think I've got a formula that splits the interval up in 4n pieces of length h. I've inserted it below (I=...)

I = (2h/45) [7f(a)+32f(a+h)+12f(a+2h)+32f(a+3h)+7f(a+4h)] + (2h/45) [7f(a+4h)+32f(a+5h)+12f(a+6h)+32f(a+7h)+7f(a+8h)] + ... + (2h/45) [7f(a+(n-4)h)+32f(a+(n-3)h)+12f(a+(n-2)h)+32f(a+(n-1)h)+7f(a+nh)]

I = (2h/45)[7f(a+(n-4)h)+32f(a+(n-3)h)+12f(a+(n-2)h)+32f(a+(n-1)h)+7f(a+nh)]

I'm not sure how I would determine the error bound formula for the composite rule though.

h = (b-a)/(4n)

Error bound formula = - (8h⁷/945)*d⁶g/dt⁶ (ξ)
for some ξ ∈ [0,4h]