Sivakumar MATH 152H
Example Sheet 1b
1. Recall that, if fis continuous on [a, b], then
Zb
a
f(x)dx = lim
n→∞
b−a
n
n
X
k=1
fa+k(b−a)
n.
Use this to evaluate the following limit:
lim
n→∞
1
√n1
√n+ 1 +1
√n+ 2 +· ·· +1
√n+n.
2. Suppose fis a continuous function and ais a fixed positive number. Prove the following
statements:
(i) If fis an odd function, then Za
−a
f(x)dx = 0.
(ii) If fis an even function, then Za
−a
f(x)dx = 2 Za
0
f(x)dx.
3. Evaluate the following integrals:
(a) Z3
0|x(x−2)|dx
(b) Zcot x dx
(c) Ze4
e
dx
x√ln x
(d) Z4
0
x
√2x+ 1 dx
(e) Zx
1 + x4dx
(f) Zx3px2+ 1 dx
(g) Zπ/2
−π/2
x2sin x
1 + x6dx
(h) Za
0
xpa2−x2dx, where ais a fixed positive number
4. If aand bare positive integers, show that
Z1
0
xa(1 −x)bdx =Z1
0
xb(1 −x)adx.
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