Math 105/206 - Quiz 2, Feb 6 2015
IMPORTANT: Write your name AND student number somewhere on this sheet.
No calculators, books or notes. Please show your work to get full marks. (10 marks total +2 “bonus” marks)
If you get more than 10 marks, the excess ones will be transferred to other quizzes.
Problem 1
Given the function f (x) = 3x − 2 :
(a) Compute the right Riemann sum
R 2of f , with n = 4 subintervals and relative to the interval [0, 2] (2 marks).
(b) Compute the definite integral 0 f (x)dx by using its interpretation as a net area (don’t use the fundamental theorem of calculus) (2 marks).
R2
(2 bonus marks) Compute the definite integral 0 f (x)dx by writing it as a limit of right Riemann sums
P
with n subintervals of equal length. You will need the formula nk=1 k = n(n+1)
. Your final result should be a
2
number (the same one you get from point (b) above, hopefully).
Problem 2
(a) Compute the indefinite integrals (3 marks)
Z
Z
3x3 − 4x2 + 1
dx
(cos(x) + 2 sin(3x))dx
x2
Z
e4x dx.
(b) Use the fundamental theorem of calculus and the results of (a) to compute the definite integrals (3
marks)
Z 1
Z 2π
Z 2 3
3x − 4x2 + 1
(cos(x) + 2 sin(3x))dx
e4x dx.
dxdx
2
x
π
0
1