Quiz 3 for MATH 105 SECTION 205
February 04, 2015
Given Name
Family Name
Student Number
Z
2
1. (a) (1 point) Write down the midpoint Riemann sum with n = 50 to approximate
1
notations.
1
dx using sigma
x
(a)
(b) (0.5 points) Express 2 + 4 + 6 + 8 + · · · + 80 using sigma notations.
(b)
Z
(c) (1 point) Let f and g be two integrable functions on [1, 2], if
Z 2
compute
[2f (x) − 3g(x)] dx.
2
Z
2
g(x) dx = 1,
f (x) dx = 2 and
1
1
1
(c)
√
1 − x2 , if 0 ≤ x ≤ 1,
2. Let f (x) =
, and R be the region bounded by the graph of f (x) and x-axis between
2 − 2x,
if x > 1,
x = 0 and x = 3, then
(a) (1 point) Find the area of R.
(a)
(b) (1 point) Find the net area of R.
(b)
Z
(c) (0.5 points) Compute
3
f (x) dx.
0
(c)
3. Let f (x) =
1, if x is a rational number
, then
0, if x is not a rational number
(a) (1. points) What’s the value for the left Riemann sum for any regular partition of [0, 1]?.
(a)
(b) (0.5 points) For any partition of [0, 1], is it true that we can always find a Riemann sum whose value is
0? (Just put ‘Yes’ or ‘No’).
(b)
(c) (0.5 points) Is f integrable on [0, 1]? (Just put ‘Yes’ or ‘No’).
(c)
Z
4. (3 points) Use the Riemann sum to compute
1
(2x + 1) dx.
0
Your Score:
/10