The Fundamental Theorem of Calculus and the Definite Integral

1
The Fundamental Theorem of Calculus and the
Definite Integral

• Section 6.4 - 6.5

2
Accumulation Functions
Workers Addl Output
1 3
2 6
3 9
4 12
5 15
Workers Output
1 3
2 9
3 18
4 30
5 45
3
Accumulation Functions
4
Accumulation Functions
5
Accumulation Functions
6
Accumulation Functions
Midpoint Approximation
7
Accumulation Functions
Midpoint Approximation
8
Accumulation Functions
Midpoint Approximation
9
Accumulation Functions
Midpoint Approximation
10
Accumulation Functions
11
Fundamental Theorem of Calculus

• Fundamental Theorem of Calculus

Differentiation and integration reverse each
other
12
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
13
Fundamental Theorem of Calculus
14
Fundamental Theorem of Calculus

• Total number of customers
• Between 9am and 9pm
• Number of customers per minute
• Sample throughout day

15
Accumulation Functions
Time Customers Time Customers
900 1 230 5
945 2 310 5
1015 3 400 4
1100 4 515 4
1145 4 630 3
1215 5 730 2
115 5 815 2
16
Accumulation Functions
17
Accumulation Functions

• Fit a quadratic model to the data

18
Fundamental Theorem of Calculus
Total number of customers Area between curve
and x-axis
19
Fundamental Theorem of Calculus
Y
Y1
CALC
7 f(x)dx
20
Fundamental Theorem of Calculus

• In-Class