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NATIONAL UNIVERSITY OF SINGAPORE SEMESTER 1, 2021/2022 MA2002 Calculus Homework Assignment 4 IMPORTANT: (i) Write your name and student number on every page of your answer scripts. (ii) Scan your scripts as a single PDF document. Other formats are not acceptable. (iii) Rename your PDF document by student number. For example, A1234567X.pdf. (iv) Log in to LumiNUS, and upload your PDF document to one of the subfolders in Files — Student Submission — Homework Assignment 4. (v) Submit by 8th November 2021 (Monday) 23:59. 1. Use Riemann sums to evaluate the following definite integrals. [8] (a) ∫ 3 1 2x d x. (b) ∫ 6 0 cos3x d x. [Hint: n∑ i=0 cos(iθ) = 1 2 + sin( 2n+1 2 θ) 2sin θ2 ]. 2. Express the following as definite integrals and evaluate the limits. [10] (a) lim n→∞ ( 1 n5 n∑ i=1 i 3 √ n2 − i 2 ) . (b) lim n→∞ n∑ i=1 n3 16n4 − i 4 . 3. Find the following limits. [8] (a) lim x→0 ( cos x cos3x )csc2 x . (b) lim x→∞ ( x2 +px +1 x2 −px +1 )x3/2 . 4. Find the following definite integrals. [8] (a) ∫ 3 1 d x (x +3)2 p x2 +2x −3 . [Hint: Trigonometric substitution.] (b) ∫ 1 0 √ 1−x2 sin−1 x d x. [Hint: Integration by parts.] 1 MA2002 CALCULUS HOMEWORK ASSIGNMENT 4 2 5. Let m,n be nonnegative integers and define I (m,n) = ∫ 1 0 xm(ln x)n d x. Find a relation between I (m,n) and I (m,n −1) (n ≥ 1) and find a general formula of I (m,n). [6] [Hint: This is an improper integral.] 6. Let f be a continuous function on [a,b]. If ∫ b a [ f (x)]2 d x = 0, prove that f (x) = 0 for all x ∈ [a,b]. [5] [Hint: Prove by contradiction. The precise definition of limit may be necessary.] 7. Let f and g be continuous increasing functions on [0,1]. Prove that [5]∫ 1 0 f (x)d x ∫ 1 0 g (x)d x ≤ ∫ 1 0 f (x)g (x)d x. [Hint: Use mean value theorem for definite integral.]