January 24, 2006 Lecturer: Dr Martin Kurth Hilary Term 2006 Course 1E1 2004-2005 (JF Engineers & JF MSISS & JF MEMS) Problem Sheet 2 Due: in the Tutorials 03 February / 06 February The Mean Value Theorem is the midwife of calculus – not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance. from: E. Purcell and D. Varberg Calculus with Analytic Geometry 1. State both parts of the Fundamental Theorem of Calculus. (4 points) 2. Calculate the following definite integrals. (a) Z 2 1 (x + 3)dx 0 (b) Z 1 (x + 2)2 dx 0 (c) − Z √π/2 x sin(x2 )dx 0 (6 points) 3. Without using integration by parts (which we will learn about later), show that Z π π sin2 (x)dx = . 2 0 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.