MIT 18.01 Single Variable Calculus: Problem Set 8A, Study notes of Mathematics

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18.01 Single Variable Calculus

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18.01 Problem Set 8A

Due Friday 12/08/06, 1:55 pm 8A is the first half of Problem Set 8, all of which is due a week after Exam 4. (The second half, 8B, will be issued at the exam.) Even though it won’t be collected until later, you should do 8A before the exam, to prepare for it. Part I (15 points) Lecture 30. Fri. Nov. 17 Parametric equations; arclength. Surface area. Read 17.1, 7.5, 7.6 Work: 4E-2, 3, 8; 4F-1d, 4, 5, 8; 4G-2, 5. If a curve is given by x = x(t), y = y(t), to find its arclength, use ds in the form ds = � (dx)^2 + (dy)^2 = � dx � 2

� dy � 2 dt , dt dt and integrate from start to finish: from t = t 0 to t = t 1. Lecture 31. Tues. Nov. 21 Polar coordinates; area in polar coordinates. Read: 16.1, (16.2 lightly, for the pictures), 16.3 to top p.570, 16.5 to middle p. Work: 4H-1bfg; 4H-2a,3f; 4I-2, Lecture 32. Tues. Nov 28. Continuation and review. Lecture 33. Thurs. Nov 30. Exam 4, 1:05-1:55, lectures 25–32. Part II (30 points) Directions: Attempt to solve each part of each problem yourself. If you collaborate, solutions must be written up independently. It is illegal to consult materials from previous semesters. With each problem is the day it can be done.

1. (not until due date; 3 pts) Write the names of all the people you consulted or with whom you collaborated and the resources you used, or say “none” or “no consultation”. (See full explanation on PS1).
2. (Lec 30, 7 pts: 3 + 1 + 3) a) Find the algebraic equation in x and y for the curve x = a cosk^ t, y = a sink^ t. Draw the portion of the curve 0 ≤ t ≤ π/ 2 in the three cases k = 1, k = 2, k = 3. b) Without calculation, find the arclength in the cases k = 1 and k = 2. c) Find a definite integral formula for the length of the curve for general k. Then evaluate the integral in the three cases k = 1, k = 2, and k = 3. (Your answer in the first two cases should match what you found in part (b), but the calculation takes more time.)
3. (Lec 30, 9 pts: 3 + 1 + 3 + 2) The hyperbolic sine and cosine are defined by cosh x = ex^ + e−x , sinh x = ex^ − e−x 2 2 1