1. Exam II: Wed, November 3.
Review session: Mon, November 1.
SI Review session: Tue, November 2.
Topics: FTC 5.3,
Net Change Th 5.4,
substitution rule 5.5,
integration by parts 7.1,
trigonometric integrals 7.2,
trigonometric substitution 7.3,
integration of rational functions 7.4,
areas between curves 6.1.

3. Problems
1. FTC, Part 1: Differentiation of integrals.
Sec
2. Net change Th.
Motion in a straight line. Given velocity function v(t), find displacement and total distance.
Sec
3. Substitution
indefinite integrals:
definite integrals:
Sec
4. By parts:
Sec , 63-64
5. Trigonometric integrals:
Sec
6. Trigonometric substitution:
Sec
etc.

4. 7. Rational functions:
Sec
Completing square:
Sec
8. Area between curves
Set up the area between the following curves: a) b)
Sec

5. General strategy for integration: some recommendations
Simplify the integrand;
Try obvious substitution;
Classify the integrand according to its form:
Trigonometric f-n 
try substitution
as in Sec. 7.2
(if applicable)
Product of f-ns 
try by parts
Rational f-n 
try partial fractions
Radicals
if if
 
try try
trigonometric rational
substitution substitution
as in Sec. 7.3
4. Manipulate the integrand (e.g. algebraic, trigonometric manipulations);
5. Combine different methods;
6. Be creative!!!

6. Areas between curves S x y a y=f(x) b y=g(x)
Consider region S between y=f(x), y=g(x), x=a, x=b,
where f, g – continuous, f(x)g(x) x[a,b].
A – area of region S x y a
y=f(x) b
y=g(x)
Divide [a,b] into n subintervals of width
Approximate an area of i-th strip by
a rectangle with base x and height
where is some sample point in the interval [xi,xi+1].
Then

7. y S2 x a
y=f(x) b
y=g(x) S1
f(x)g(x) for some x[a,b]
g(x)f(x) for some x[a,b]
Then split S into sub-regions and find an area of each.
Total area is S x y c
x=f(y) d
x=g(y)
Consider region S between x=f(y), x=g(y), y=c, y=d,
where f, g – continuous, f(y)g(y) y[c,d].
Then