1) Set up the linear functional

Simpson’s rule on [a,b][a,b][a,b] (with midpoint m=a+b2m=\tfrac{a+b}{2}m=2a+b​) is

Q(f)=b−a6[f(a)+4f(m)+f(b)].Q(f)=\frac{b-a}{6}\big[f(a)+4f(m)+f(b)\big].Q(f)=6b−a​[f(a)+4f(m)+f(b)].

Define the error functional

L(f) := ∫abf(x) dx−Q(f).L(f)\;:=\;\int_a^b f(x)\,dx - Q(f).L(f):=∫ab​f(x)dx−Q(f).

Simpson is exact for all polynomials of degree ≤3\le 3≤3. Therefore, the

Peano Kernel Theorem applies with order r=4r=4r=4:

L(f)=∫abK(t) f(4)(t) dt,L(f)=\int_a^b K(t)\,f^{(4)}(t)\,dt,L(f)=∫ab​K(t)f(4)(t)dt,

where the kernel is

K(t)=L ⁣((⋅−t)+ 3)3!with(x−t)+:=max⁡{x−t,0}.K(t)=\frac{L\!\big((\cdot-t)_+^{\,3}\big)}{3!} \quad\text{with}\quad (x-t)_+ := \max\{x-t,0\}.K(t)=3!L((⋅−t)+3​)​with(x−t)+​:=max{x−t,0}.

2) Compute L ⁣((⋅−t)+ 3)L\!\big((\cdot-t)_+^{\,3}\big)L((⋅−t)+3​)

Let ϕt(x)=(x−t)+3\phi_t(x)=(x-t)_+^{3}ϕt​(x)=(x−t)+3​. Then:

(i) The integral part)

∫abϕt(x) dx=∫tb(x−t)3 dx=(b−t)44.\int_a^b \phi_t(x)\,dx = \int_t^b (x-t)^3\,dx = \frac{(b-t)^4}{4}.∫ab​ϕt​(x)dx=∫tb​(x−t)3dx=4(b−t)4​.

(ii) The quadrature part)

Q(ϕt)=b−a6[ϕt(a)+4ϕt(m)+ϕt(b)].Q(\phi_t) = \frac{b-a}{6}\Big[\phi_t(a)+4\phi_t(m)+\phi_t(b)\Big].Q(ϕt​)=6b−a​[ϕt​(a)+4ϕt​(m)+ϕt​(b)].

Because ϕt(x)=0\phi_t(x)=0ϕt​(x)=0 for x≤tx\le tx≤t:

• ϕt(a)=0\phi_t(a)=0ϕt​(a)=0 for all t∈[a,b]t\in[a,b]t∈[a,b].
• ϕt(m)=(m−t)3\phi_t(m)=(m-t)^3ϕt​(m)=(m−t)3 if t≤mt\le mt≤m; otherwise 000.
• ϕt(b)=(b−t)3\phi_t(b)=(b-t)^3ϕt​(b)=(b−t)3 for all t∈[a,b]t\in[a,b]t∈[a,b].

Hence,

Q(ϕt)=b−a6[4 [m−t]+3+(b−t)3],[⋅]+:=max⁡{⋅,0}.Q(\phi_t)= \frac{b-a}{6}\Big[4\,[m-t]_+^{3}+(b-t)^3\Big], \quad [\cdot]_+ := \max\{\cdot,0\}.Q(ϕt​)=6b−a​[4[m−t]+3​+(b−t)3],[⋅]+​:=max{⋅,0}.

Putting (i) and (ii) together and dividing by 3!=63!=63!=6, we get the Peano kernel:

K(t)=16[(b−t)44−b−a6(4 [m−t]+3+(b−t)3)].\boxed{ K(t) = \frac{1}{6}\left[ \frac{(b-t)^4}{4} - \frac{b-a}{6}\Big(4\,[m-t]_+^{3}+(b-t)^3\Big) \right]. }K(t)=61​[4(b−t)4​−6b−a​(4[m−t]+3​+(b−t)3)].​

This is equivalently the following piecewise cubic (since [m−t]+3[m-t]_+^3[m−t]+3​ is piecewise):

K(t)={16 ⁣[(b−t)44−b−a6(4(m−t)3+(b−t)3)],a≤t≤m,16 ⁣[(b−t)44−b−a6(b−t)3],m