February 7, 2006 Lecturer: Dr Martin Kurth Hilary Term 2006 Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS) Problem Sheet 4 Due: in the Tutorials 17 February / 20 February To demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. Jules Henri Poincaré (1854–1912) 1. Calculate the volume of the solid generated by revolving the area between the x-axis and the graph of the function f with f (x) = 1 − x2 about the line x = −2. (5 points) 2. Calculate the length of the graph of f (x) = x2/3 between x = 1 and x = 8. (5 points) 3. Show that the shortest smooth curve connecting two points (x1 , y1 ) and (x2 , y2 ) is a straight line. (*) Questions 1 and 2 should be answered by all students, you will get points for them. Question 3 is more challenging and meant as an exercise for the more mathematically interested students.