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Math1014 Calculus II
Week 9: Brief Review and Some Practice Problems
Improper Integrals
• Evaluating improper integrals by taking suitable limits of ordinary integrals.
• Determining convergence or divergence of improper integrals by comparison.
1. Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
Z 1
Z 0
Z ∞
1
x
e−2t dt
dx
(iii)
(i)
dx
(ii)
2 + 2)2
2x
−
5
(x
−∞
−∞
0
Z ∞
Z 6
Z 3
1
1
√
(iv)
dx
dz
(v)
rer/3 dr
(vi)
2 + 3z + 2
z
3
−x
0
−∞
2
Z 8
Z 2 1/x
Z 1
1
e
ln x
√ dx
(vii)
dx
(viii)
dx
(ix)
3
3
(x
−
6)
x
x
6
0
0
2. The average speed of molecules in an ideal gas is
Z
4 M 3/2 ∞ 3 −M v2 /(2RT )
v¯ = √
v e
dv
π 2RT
0
where M is the molecular weight of the
r gas, R is the gas constant, T is the temperature, and v is
8RT
.
the molecular speed. Show that v¯ =
πM
3. Find the value of the constant C for which the integral
Z ∞
x
C
−
dx
x2 + 1 3x + 1
0
converges. Evaluate the integral for this value of C.
4. Show that if a > −1, and b > a + 1, then the following integral is convergent.
Z ∞
xa
dx
1 + xb
0