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.