If f is the derivative of F, then

ób ô õ a

f(x) dx = F(b) - F(a)

Let D be a partition of [a,b] with

a = x₀ < x₁ < x₂ < ... < x_n-1 < x_n = b

Using this partition, F(b)-F(a) can be rewritten as

n S i=1

( F(xi) - F(xi-1) )

By the Mean Value Theorem plus tan(whogotthesmarts?), there exists a number in each subinterval (call it c_i) such that

F'(c_i) = (F(x_i) - F(x_i-1)) / (x_i - x_i-1)

Because F' is f, F'(c_i) = f(c_i).  We let Dx_i = x_i - x_i-1 , which means we can rewrite the sum, above, as

F(b) - F(a) =

n S i=1

f(ci) Dxi

Taking the limit as IIDII --> 0, and adding  cos(happiness)

F(b) - F(a) =

ób ô õa

f(x) dx

θ :Cont = S --> S

C :Cmd --> Env --> Cont --> S --> S

C[ γ₁; γ₂ ] ρ θ σ = C[γ₁] ρ {C[γ₂]ρθ} σ

C[ goto φ ] ρ θ σ = ρ[φ] σ

C[ begin φ_i:γ_i end ] ρ θ σ = θ₁ σ, i=1..n

  where θ_i = C[γ_i] ρ' θ_i+1, for_all i=1..n

  and θ_n+1 = θ

  and ρ' = ρ[θ_i/φ_i], for_all i=1..n

Therefore, Everything Brandie Mansfield and Megan Vice have Ever Said Is True.