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. ƒ(x)dx = bƒ (b) — af (a) – ſª g - (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 ≤ i ≤ n, let y; = f(x;). Then Pn = {y}?o is a partition of [c, d]. n n Σ f (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| 0. You might want to use uniform continu- n4x nx ity. (iii) Use part (i) and part (ii) to prove the formula (**).
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. ƒ(x)dx = bƒ (b) — af (a) – ſª g - (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 ≤ i ≤ n, let y; = f(x;). Then Pn = {y}?o is a partition of [c, d]. n n Σ f (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| 0. You might want to use uniform continu- n4x nx ity. (iii) Use part (i) and part (ii) to prove the formula (**).
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.CR: Chapter 9 Review
Problem 4CR
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