1. Calculus & Exam Section 6

2. Methods
One can calculate derivatives and integrals via two different methods:
Analytically – Lots of scary math.
Graphically – Some drawing and some not-so-scary math.

3. Graphical Integration
In order to do graphical integration:
Split up the area under the curve into small rectangles.
Calculate the area of each of the rectangles.
Sum all the areas of the rectangles.

4. Graphical Integration Example
Integrate the function from x = 0 to x = 4:
area = = 10.5
2.000
2.000
1.625
1.625
1.25
1.25
.375
.375
The actual area under the curve is 8 when calculated analytically. The more rectangles you use the better your precision.

5. On to the exam problem…

6. The Problem
You were asked to use graphical integration to graph a stock with the following inflows and outflows assuming the initial value of the stock to be 100: 5 10 15 20 40 80
Flows (units/time)
Inflow
Outflow

7. First Things First
First, calculate the net flow by graphically subtracting the outflow from the inflow: 80
Inflow
Outflow
Flows (units/time) 40
Net Flow
60 – 50 = 10
60 – 80 = -20
60 – 70 = -10
30 – 40 = -10
50 – 40 = 10
70 – 40 = 30
40 – 30 = 10
60 – 30 = 30
20 – 20 = 0
20 – 40 = -20
-20 5 10 15 20

8. 0 ≤ time ≤ 2.5 Consider the region 0 ≤ time ≤ 2.5 (highlighted):80
Inflow
Outflow
Flows (units/time) 40
Net Flow
-20 5 10 15 20
Time
1.25
2.50
Net Flow
-20
Calculation
20 – 20 = 0
= -20
Note that the outflow function is discontinuous at time = 2.5, i.e. it has two values, flow = 20 and flow = 40. Thus, the net flow function also has two values, net flow = 0 and net flow = - 20.

9. 2.5 < time ≤ 10
Consider the region 2.5 < time ≤ 10 (highlighted): 80
Inflow
Outflow
Flows (units/time) 40
Net Flow
-20 10 15 20
Time
3.75
5.00
6.25
7.5
8.75 10
Net Flow
-20
-7.5 5
17.5 30
Calculation
45-40
40-30
Note that both the outflow and the inflow function are discontinuous at time = 10. The left limit (the value that the function approaches as it moves from left to right) is 70 for the inflow function and 40 for the outflow function. 70 – 40 = 30 is the left limit of the net flow function. The right limit (the value that the function approaches as it moves from right to left) is 40 for the inflow function and 30 for the outflow. 40 – 30 = 10 is the right limit of the net flow function.

10. 10 < time ≤ 20
The following is the solution for the remaining points of the net flow: 80
Inflow
Outflow
Flows (units/time) 40
Net Flow
-20 10 15 20
Time
12.50
15.00
16.25
17.5
18.75 20
Net Flow 10 30 5
-7.5
-20
Calculation
40 – 30
60 – 30
60 – 67.5

11. Integration
Now, chop up the Net Flow function with rectangles and calculate the area for each: 80
Flows (units/time) 40
Net Flow 31
1.25 x 25 = 31.25 31
1.25 x 25 = 31.25
1.25 x 0 = 0
1.25 x 0 = 0
1.25 x 0 = 0 16
1.25 x 12.5 = 15.63 16
1.25 x 12.5 = 15.63
1.25 x 0 = 0 12
1.25 x 10 = 12.5 12
1.25 x 10 = 12.5 12
1.25 x 10 = 12.5 12
1.25 x 10 = 12.5
-25
1.25 x -20 = -25
-25
1.25 x -20 = -25
-16
1.25 x 12.5 = 15.63
-16
1.25 x =
-20 5 10 15 20

12. Graphing
The next step is to calculate the y-value for the given point by adding the area of the integral (area of the rectangle) to the y-value of the previous point.
For example, for x = 1:
y = = 100
and so on…

13. Graphing Step By Step 80 Net Flow Flows (units/time) 40 32 31 0 0 16
0 0 16 16 12 13 12 13
-25
-25
-16
-16
-20 5 10 15 20
= 118
= 130
= 161
= 177
= 100
= 100
= 106
= 177
= 161
= 75
= 50
= 82
= 94
= 50
= 34
= 34
100 5 10 15 20
100
200
Stock (units)

14. Analytical Solution
The following is the graph of the analytical solution… for the math geek in all of us:
y = 100
y = 10x - 25
y = -20x + 150
y = 5x2 – 70 x + 275
y = 5x x -1450