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MATH 101 HOMEWORK 6 Due on Wednesday Oct. 27, 2004. For full credit, show all work. Calculators are not allowed, except as indicated below. 1. (8 marks) Evaluate the integrals, or show that they diverge: Z ∞ ln x (a) dx, x3 1 Z 1 ln x √ . (b) x 0 2. (8 marks) Use the comparison test to decide whether the following improper integrals are convergent. Justify your answer. Z 2 2 − sin x (a) dx, 2−x 0 Z π dx √ (b) . sin x 0 R1 3. (4 marks) Write out the midpoint approximation Mn to the integral 0 sin(x2 )dx. If we want the error to be no greater than .001, how large should n be? In this problem, you may use a calculator. This homework covers Sections 6.5–6.6 of the textbook. Additional recommended exercises: Section 6.5: 1–41; Section 6.6: 1–11. 1