SCIE 001 MATHEMATICS ASSIGNMENT 9 (Due Jan. 23, 2013) There are two parts to this assignment. The first part is on WeBWorK and is due by 10:00 am on Wed. Jan. 23. The second part consists of the questions on this page, and is due before 10:00 am on Wed. Jan. 23. For these questions, you are expected to provide full solutions with complete arguments and justifications. You will be graded primarily on the correctness, clarity and elegance of your solutions. Your answers must be typeset or very neatly written. They must be stapled, with your name and student number at the top of each page. 1. (a) Solve the integral equation Z 3 f (x) = 1 + 2 f (t) dt x for f (x); that is, find an expression for f (x) in terms of elementary functions. Z 2π √ (b) Find cos x − 3 sin x dx. 0 Z yq 1 2 (c) Bonus: Let y ∈ [0, 1]. Prove that x4 + (y − y 2 ) dx ≤ . 3 0 2. There is a line y = mx through the origin that divides the finite region, bounded by the curve y = x − x2 and the x-axis y = 0, into two regions with equal area. Find the slope m of the line. 3. See p. 467, Example 6, for an example of a reduction formula. Use integration by parts to prove the reduction formula Z Z tan x secn−2 x n − 2 n sec x dx = + secn−2 x dx n−1 n−1 R R for any integer n ≥ 2, and use this reduction formula to find sec3 x dx and sec5 x dx.