Integration (2 Viewers)

[Xayma]

Can you repost it, if I can do part of it with my knowledge it would probably be realtively earlier.

 

[Grey Council]

will do at home

 

[clive]

Originally posted by Orange Council
umm, did someone give any thought to the question i posted up earlier?

when you hafta prove that one curve is higher than another curve, and then the question asks you to prove that the integral of one of the curves is more than a certain number but less than another.
Inequations and integration, i think the topic is.

where abouts in a paper would that appear?

Have you looked at 2002 HSC question 6 (b) (i think)?. That involved some integrals and inequalities.

 

[Grey Council]

yes, 2002 HSC question 6 (b) is the type of questions i'm referring to.

so i think that answers my question. ^_^
its around question 6 in the paper.

ta clive

 

[CM_Tutor]

Orange Council, those types of questions tend to be nearer the ends of papers, and here are a couple for people to think about:

1. Consider the graph of y = 1 / t, for t > 0.

(a) Use a diagram to show that int (from 1 to sqrt(x)) 1 / t dt < sqrt(x) for x > 1

(b) Hence, or otherwise, prove that lim_{x--> + inf} (ln x) / x = 0

(c) Using appropriate substitutions to show that
(i) lim_x-->0⁺ xln x = 0
(ii) lim_{x--> - inf} xe^x = 0

(d) By construct a similar proof to that found in (a) and (b), or otherwise, show that lim_x-->0⁺ x^aln x = 0 for any real a > 0.

2. Draw a large diagram of y = ln x, and on it mark the points A(1, 0), B(2, ln 2), C(3, ln 3), D(4, ln 4), Y(n, ln n) and X((n - 1), ln(n - 1)), where n is an integer and n > 1. Join A to B, B to C, C to D and X to Y. By comparing areas, show that n! < e.n^n+1/2 / eⁿ for integers n > 1

3. Draw a large diagram of y = x³ for x => 0. Mark on the curve the points P₁, P₂, P₃, ..., P_n, where P_k has x coordinate ka / n, given a is a positive constant. The feet of the perpendiculars from P₁, P₂, P₃, ... P_n to the x- and y- axes meet those axes at X₁, X₂, X₃, ... X_n and Y₁, Y₂, Y₃, ... Y_n, respectively. The points L₁, L₂, L₃, ... L_n-1 are located such that L_k, at ((k + 1)a / n, (ka / n)³), is the intersection of X_k+1P_k+1 and Y_kP_k produced. The points U₁, U₂, U₃, ... U_n-1 are located such that U_k is at the intersection of X_kP_k produced and Y_k+1P_k+1. O is the origin.

(a) Draw a diagram to represent this information.

NOTE: For the rest of this question, you may use (without proof) the result 1³ + 2³ + 3³ + ... + n³ = n²(n + 1)² / 4

(b) Show that the sum of the areas of the lower rectangles shown in the diagram (the lower rectangles are X₁X₂L₁P₁, X₂X₃L₂P₂, ..., X_n-1X_nL_n-1P_n-1) is a⁴(n - 1)² / n².

(c) Find an expression for the sum of the areas of the upper rectangles shown in the diagram (the upper rectangles are OX₁P₁Y₁, X₁X₂P₂U₁, X₂X₃P₃U₂, ..., X_n-1X_nP_nU_n-1).

(c) Hence, show that (a⁴ / 4) * (1 - 1 / n)² < int (from 0 to a) x³ dx < (a⁴ / 4) * (1 + 1 / n)², and prove (without using integration) that int (from 0 to a) x³ dx = a⁴ / 4.