Riemann Integrability ·f∈C[a,bl,[a,b]bdd→f is Riemann integrable.·When integrable,.and max subinterval in P→0(IPl→0)： b lim L(f,P)= f(x)da =lim U(f,P) 1P→0 |P→0 Counter example:Dirichlet function 0,x rational, d()1,irrational →L=0,U=b-a Copyright©2011NA⊙Yin 12
Riemann Integrability • f ∈ C 0 [a, b], [a, b] bdd ⇒ f is Riemann integrable. • When integrable, and max subinterval in P → 0 (|P| → 0): lim |P |→0 L(f, P) = Z b a f(x)dx = lim |P |→0 U(f, P) • Counter example: Dirichlet function d(x) ≡ 0, x rational, 1, x irrational ⇒ L = 0, U = b − a Copyright c 2011 NA Yin 12
Challenge:Estimate n for Third Method Current restrictions for n estimate: Monotone functions Uniform partition ●Challenge: *estimate ecosd rπ error tolerance=x 10-3 using L and U *几=？ Copyright©2011NA⊙Yin 13
Challenge: Estimate n for Third Method • Current restrictions for n estimate: ∗ Monotone functions ∗ Uniform partition • Challenge: ∗ estimate R π 0 e cos xdx ∗ error tolerance = 1 2 × 10−3 ∗ using L and U ∗ n =? Copyright c 2011 NA Yin 13
Estimate n-Solution .f(x)=ecosa on [0,T]..mi=f(xi+1)and Mi=f(xi) ·.L(f:P)=h∑f(ci+)andU(f:P)=h∑f(x,h=开 ●Want(U-L)<3×10-3or(e2-e-1)<10-3 ....n >7385(!!)(note for later:max error estimate =O(h)) Number of f(x)evaluations 2 for (U-L)max error calculation *>7000 for either L or U. We need something better Copyright 2011 NAOYin 14
Estimate n – Solution • f(x) = e cos x & on [0, π] ∴ mi = f(xi+1) and Mi = f(xi) • ∴ L(f : P) = h Pn−1 i=0 f(xi+1) and U(f : P) = h Pn−1 i=0 f(xi), h = π n • Want 1 2 (U − L) < 1 2 × 10−3 or π n (e 1 − e −1 ) < 10−3 • . . . n ≥ 7385(!!) (note for later: max error estimate = O(h)) • Number of f(x) evaluations ∗ 2 for (U − L) max error calculation ∗ > 7000 for either L or U. We need something better Copyright c 2011 NA Yin 14