Applications of the Definite Integral

Abstract

Semester I, 2016-17 1. Claim. Let f : [a, b] → IR be continuous. Suppose that for any closed subinterval [α, β] of [a, b], a real valued quantity Q(α, β) satisfies the following axioms: A1. Let m, M be, respectively, the minimum and maximum of f on [α, β] m(β − α) ≤ Q(α, β) ≤ M (β − α) A2. For any γ with α < γ < β Q(α, β) = Q(α, γ) + Q(γ, β) then Q(a, b) = b a f (1) Proof. Let P = {a = x 0 < x 1 < x n = b} be a partition of [a, b]. Let m i and M i denote the minimum and maximum respectively of f on the subinterval [x i−1 , x i ], 1 ≤ i ≤ n. Then, m i (x i − x i−1) ≤ Q(x i−1 , x i) ≤ M i (x i − x i−1) Summing over all i yields: L(f, P) ≤ n i=1 Q(x i−1 , x i) ≤ U (f, P) Repeated application of axiom [A2] (or induction) gives n i=1 Q(x i−1 , x i) = Q(a, b) Equation (1), follows from the fact that the integral is the unique number which lies between every upper sum and every lower sum. 2. Example. Let f : [a, b] → IR be non negative. If, for every subinterval [α, β] ⊆ [a, b], it is required that the area A(α, β) under the graph of f between x = α and x = β satisfy [A1] and [A2] then necessarily: A(a, b) = b a f 3. Example Let x(t), t ∈ [a, b] be the position at time t of a particle in rectilinear motion. If Define m = min{|x (t)| : α ≤ t ≤ β} M = max{|x (t)| : α ≤ t ≤ β} then the distance travelled L = L(a, b) in the time interval α ≤ t ≤ β is required to satisfy (a) m(β − α) ≤ L(α, β) ≤ M (β − α) (b) For any γ with α < γ