Student number
Name [SURNAME(S), Givenname(s)]
MATH 101, Section 212 (CSP)
Week 1: Marked Homework Assignment
Due: Thu 2011 Jan 13 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Let f (x) = x − x2 , 0 ≤ x ≤ 2, and divide the interval [0, 2] into n subintervals of equal
width.
(a) Find the Riemann sum Rn taking the sample points to be right endpoints of each
subinterval [xi−1 , xi ].
(b) Find the Riemann sum Ln taking the sample points to be left endpoints of each
subinterval [xi−1 , xi ].
(c) Find the Riemann sum Mn taking the sample points to be midpoints of each
subinterval [xi−1 , xi ].
In each case (a), (b) and (c) find i) the expression for general n, ii) the value for n = 10,
iii) the value for n = 100, and iv) the limiting value as n → ∞. For ii)–iv) give the
answers both as exact fractions and as decimals accurate to four decimal places.
2. For each of the following limits, i) express the limit as a definite integral, and ii) sketch
a region whose area is equal to the value of the limit. Do not evaluate the limits.
m
X
n
1 X
i
+1
(a) lim 2
i
n→∞ n
n
i=1
k
k2
(b) lim
cos
2
m→∞
m2
k=1 m
!
3. Evaluate the definite integral by interpreting it in terms of areas.
(a)
Z
√
2
−2
4−
x2
dx
(b)
Z
3
−1
(3 − 2u) du
(c)
Z
10
0
|y − 5| dy
4. Without attempting to evaluate the integrals, prove the following:
√
Z 1√
Z π/4
√
π
2
π
(a) 1 ≤
1 + t2 dt ≤ 2
(b)
cos y dy ≤
≤
4
2
0
−π/4
(c)
Z
6
4
1
dx ≤
x
Z
6
4
1
dx
8−x
(d)
2
≤
5
Z
2
0
1
dv ≤ 2
1 + v2