February 14, 2006
Lecturer: Dr Martin Kurth
Hilary Term 2006
Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS)
Problem Sheet 5
Due: in the Tutorials 24 February / 27 February
Everything should be made as simple as possible, but not simpler.
Albert Einstein (1879–1955)
1. Calculate the volume of the solid generated by rotating the area bounded
by the graphs of the functions f and g defined by
f (x) =
and
1
x
1
g(x) = − (x − 3)
2
about the x-axis.
(5 points)
2. Calculate the volume of the solid generated by rotating the area bounded
by the graphs of the functions f and g defined by
f (x) =
and
1
x
1
g(x) = − (x − 3)
2
about the line x = 1/2.
(5 points)
3. Prove the following statement: If x is a real number with x ≥ 1, then
ln x ≤ x − 1.
(*)
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.