Fall 2011
MA 16200
Study Guide - Exam # 2
(1) Integration via Partial Fractions: Use for (proper) rational functions R(x)
Q(x);
If degree R(x)≥degree Q(x), i.e. rational function is improper, then do long division before
using partial fractions.
(2) Integration via Clever Substitutions/Using Integral Tables: Use a substitution to transform
integral into a form to be able to use another integral techniques (Substitution Method,
Integration by Parts, Trig Integrals, Trig Subsitution Method, etc) or use inte-
gral tables.
(3) Approximating definite integrals Zb
a
f(x)dx.
Let ∆x=b−a
n, xk=a+k∆xand xk=1
2(xk−1+xk) (Note that x0=aand xn=b)
(a) Midpoint Rule:Zb
a
f(x)dx ≈Mn= (∆x) [f(x1) + f(x2) + ···+f(xn)]
(b) Trapezoidal Rule:Zb
a
f(x)dx ≈Tn=∆x
2[f(x0) + 2f(x2) + 2f(x3) + ···+ 2f(xn−1) + f(xn)]
(c) Simpson’s Rule: Only works for neven.
Zb
a
f(x)dx ≈Sn=∆x
3[f(x0) + 4f(x2) + 2f(x3) + ···+ 2f(xn−2) + 4f(xn−1) + f(xn)]
(4) Improper integrals: Type I (unbounded intervals) Z∞
a
f(x)dx , Zb
−∞
f(x)dx or Z∞
−∞
f(x)dx;
Improper integrals of Type II (discontinuous integrand at one or both endpoints) Zb
a
f(x)dx.
Comparison Theorem: Let f(x) and g(x) be continuous for x≥a.
(a) If 0 ≤f(x)≤g(x) for x≥aand Z∞
a
g(x)dx converges =⇒Z∞
a
f(x)dx also converges.
(b) If 0 ≤g(x)≤f(x) for x≥aand Z∞
a
g(x)dx diverges =⇒Z∞
a
f(x)dx also diverges.
(5) Arc length L=Zb
aq1 + (f′(x))2dx or L=Zd
cq1 + (g′(y))2dy.
(6) Surface area of revolution: S=Z2π{ribbon radius}ds or S=Z2πr ds,
where ds =q1 + (f′(x))2dx or ds =q1 + (g′(y))2dy.
x
y
x
y
y = f(x)
ds ds
r
rx = g(y)
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