1. Definite Integrals

2. b
lim
f(x) dx = n [ f(x1) x1 + f(x2) x2 + ..... + f(xn) xn ]
a
f
b
Area = f(x) dx =
a
a b

3. 4
Given the integral (x + 1) dx =
1
Estimate the integral using left and right sums with 50
subdivisions.
left hand = 10.41
right hand = 10.59
average = 10.50

4. 4
Given the integral (x + 1) dx =
1
Draw the graph and find the exact value of the area of
the region.

5. b
Given f(x) = x approximate the definite integral f(x) dx =
a
over the interval [­2, 0].

6. In General:
­ if a function is positive­valued, the definite integral
b
f(x) dx is positive
a
­ if a function is negative­valued, the definite integral is the
negative of the area of the region between the graph
of f and the x­axis

7. 4
Evaluate: (5 ­ 2x) dx =
0
Use graphing calculator to find integral value.

8. Error bounds for Monotone Functions
a function that is either increasing or decreasing
but not both on an interval [a, b]
f
f f(b)
f(b) ­ f(a)
f(a)
a b a b

9. difference between the
Error
lower and upper sum
= f(b) ­ f(a) . ( b ­ a )
n
The only variable in this expression is the number of
subintervals, n.

10. For the function f(x) = x2 on the interval [0,2], how many
intervals of equal length must be used so that an estimate
differs from the value of
2
x2 dx by less than 0.01
0
( b ­ a ) = 2 8 < 0.01
n n n
f(b) ­ f(a) = 4 ­ 0 n > 800
4 ­ 0 .( 2 ) = 8
n n