This document provides an overview of key concepts in calculus including the definite integral, properties of definite integrals, the Mean Value Theorem for Integrals, the Average Value Theorem, the Fundamental Theorem of Calculus parts 1 and 2, and the Trapezoidal Rule for approximating definite integrals. It defines integrals, discusses how to evaluate them on a TI-84 calculator, and lists properties such as additivity, constant multiples, and order of integration. It also introduces concepts like finding the average value of a function over an interval using the integral, and relating the derivative of an integral to the original function.

1. Chapter 4
Review

2. 4.2 Do you know RAM theory?
Which over- and underapproximates increasing and
decreasing functions?
How to compute given an equation?
How to compute given data?

3. 4.3 The Definite Integral
b
Area
f(x) dx when f(x)
0
a
b
Area
f(x) dx when f(x)
0
a
Net Area:
b
f(x) dx = (area above x-axis) - (area below x-axis)
a

4. Integrals on the TI-84
b
f(x) dx
Syntax: To evaluate
a
MATH 9
fnInt(
…

5. 4.3 Properties of Definite Integrals:
a
b
f(x) dx
1. Order of Integration
f(x) dx
b
a
a
f(x) dx = 0
2. Zero
a
b
b
k f(x) dx
3. Constant Multiple
a
f(x) dx
a
b
4. Sum and Difference
k
b
[ f(x) g(x)] dx
a
b
5. Additivity
b
f(x) dx
a
c
a
c
f(x) dx + f(x) dx
a
b
g(x) dx
f(x) dx
a

6. 4.4 MVT for Integrals…
Somewhere between an inscribed and
circumscribed rectangle is a rectangle
whose area is exactly the area of the
region under the curve.
b
f(x) dx = f(c)(b a)
a

7. 4.4 Average Value Theorem
The value f(c) from the MVT is the
average value of f on [a,b]…
f(c)
1
b
(b a) a
f(x) dx

8. 4.4 Fund. Thm of Calc: Part I
b
f(x) dx =F(b)-F(a)
a

9. 4.4 Fund Thm of Calc: Part 2
x
d
[ f(t) dt] =f(x)
dx a

10. 4.6 Trapezoidal Rule
b
f(x) dx
a
b a
f(x 0 ) 2 f(x1) 2 f(x 2 )
2n
2 f(x n 1) f(x n )
where n is the number of subintervals.
The trapezoidal rule overapproximates
CCU functions and underapproximates
CCD functions.