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This question is regarding the proof of the formula rd [*° f(x)dx = bƒ(b) — aƒ(a) – [ª g(x - g(x)dx. (a) If a > 0 and c> 0, draw a figure to illustrate the formula (**). (**) (b) If f is continuously differentiable on (a, b), use integration by substitution with u = g(x) to prove the formula (**).

This question is regarding the proof of the formula rd [*° f(x)dx = bƒ(b) — aƒ(a) – [ª g(x - g(x)dx. (a) If a > 0 and c> 0, draw a figure to illustrate the formula (**). (**) (b) If f is continuously differentiable on (a, b), use integration by substitution with u = g(x) to prove the formula (**).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let f: [a, b] → R be a continuous and strictly increasing function which maps the interval [a, b] f(b). Denote by g: [c, d] → R the inverse function of f. Notice that f [a, b] → R and g: [c, d] → R are Riemann integrable. bijectively onto the interval [c, d], where c = f(a), and d = : This question is regarding the proof of the formula b rd S. f(x)dx = bf(b) — aƒ(a) – ["²9 - (a) If a > 0 and c> 0, draw a figure to illustrate the formula (**). g(x)dx. (b) If f is continuously differentiable on (a, b), use integration by substitution with u = g(x) to prove the formula (**). (i) Show that i=0 (c) Let Pn = {x}o be the regular partition of the interval [a, b] into n intervals. For 0 ≤ is n, let y; = f(x;). Then Pn = {y}?o is a partition of [c, d]. n (**) n Σ ƒ (Xi-1) (Xi − Xi−1) + Σ9(Yi) (Yi − Yi−1) = bf (b) — af (a). i=1 i=1 (ii) Show that lim |P| = 0 and lim |P| n4x nx ity. (iii) Use part (i) and part (ii) to prove the formula (**). = 0. You might want to use uniform continu-
Transcribed Image Text:Let f: [a, b] → R be a continuous and strictly increasing function which maps the interval [a, b] f(b). Denote by g: [c, d] → R the inverse function of f. Notice that f [a, b] → R and g: [c, d] → R are Riemann integrable. bijectively onto the interval [c, d], where c = f(a), and d = : This question is regarding the proof of the formula b rd S. f(x)dx = bf(b) — aƒ(a) – ["²9 - (a) If a > 0 and c> 0, draw a figure to illustrate the formula (**). g(x)dx. (b) If f is continuously differentiable on (a, b), use integration by substitution with u = g(x) to prove the formula (**). (i) Show that i=0 (c) Let Pn = {x}o be the regular partition of the interval [a, b] into n intervals. For 0 ≤ is n, let y; = f(x;). Then Pn = {y}?o is a partition of [c, d]. n (**) n Σ ƒ (Xi-1) (Xi − Xi−1) + Σ9(Yi) (Yi − Yi−1) = bf (b) — af (a). i=1 i=1 (ii) Show that lim |P| = 0 and lim |P| n4x nx ity. (iii) Use part (i) and part (ii) to prove the formula (**). = 0. You might want to use uniform continu-
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