This document discusses the concept of the definite integral and Riemann sums. It explains that if the velocity over time is not constant, the definite integral is used to find the total distance by approximating the area under the curve representing velocity versus time. This is done by dividing the interval into subintervals and taking the sum of rectangles approximating small sections of the area. As the number of subintervals increases, the Riemann sum converges to the definite integral, which precisely calculates the area and extends the concept of summation to integration.

1. 5.1 Definite Integral
Finding Distance
& Riemann Sums

2. A car travels 50 m/h for 3 hrs
time
distance
1 2 3
150
100
50
(3)(50)=150 miles
D=rt

3. A car travels 50 m/h for 3 hrs
time
velocity
1 2 3
150
100
50
(3)(50)=150 miles
D=rt
v=50 is horizontal line
150 is area

4. 0
1
2
3
1 2 3 4
If V is NOT constant find area
Under the curve to find distance.
Ex 2
1
1
8
V t
= +

5. 0
1
2
3
1 2 3 4
1 1
1
8
1
1
2
1
2
8
Ex 2
1
1
8
V t
= +
1
0
1
1
1
8
2
1
1
2
3 1
2
8
t v (rate)
D=r t
V
t

6. 0
1
2
3
1 2 3 4
1 1
1
8
1
1
2
1
2
8
Ex 2
1
1
8
V t
= +
Approximate
area:
1 1 1 3
1 1 1 2 5 5.75
8 2 8 4
+ + + = =
Left hand
Rect Approx

7. 0
1
2
3
1 2 3 4
Ex 2
1
1
8
V t
= +
Could also use
Right hand
Rect Approx
1
1
8
1
1
2
1
2
8 3
Approx
area:
1 1 1 3
1 1 2 3 7 7.75
8 2 8 4
+ + + = =

8. To approximate the area under
the curve, average the left
sum and the right sum.
5.75+7.75 = 6.75
2

9. 0
1
2
3
1 2 3 4
Ex 2
1
1
8
V t
= +
Could also use
Midpoint
Rect Approx
1.03125
1.28125
1.78125
2.53125
Approx area:
6.625

10. In this example there were
four subintervals. As the
number of subintervals
increases, so does the
accuracy.

11. 2
1
1
8
V t
= +
Approx area:
6.65624
0
1
2
3
1 2 3 4
8 subintervals:
Interval
Width is .5

12. 0
1
2
3
1 2 3 4
Inscribed
rectangles are all
below the curve:
Circumscribed
Rectangles are all
Above the curve

13. Pick an integer n. ex n = 10.
Now divide the interval
into n equal subintervals.
a b

14. Endpoints of the new Subintervals
a0, a1, a2, ..., a10
This is a partition of [a,b]

15. In each subintervals [ai -1, ai ],
pick a number xi and draw a line
segment to the x-axis from the
point (xi ,0) to a point on the
graph of the function, (xi, f(xi)).

16. The area of each rectangle
is

17. The sum of the areas of the
rectangles is
This is considered a
Riemann Sum

18. If n = 20
If n= 40

20. As the number of intervals
increases, the Riemann Sum
converges to a single number
called the definite Integral.

21. f (x)dx
a
b
ò =
lim
n®¥ f(x1)∆x1 +…. f(xn)∆xn

22. The integral is
an extension
of the concept
of a sum. The process of
finding integrals is called
integration. Integration is
used to find area under curves

23. Learn the rules for + and - Area

24. "The sum of wisdom is
that time is never lost
that is
devoted
to work." -
- Emerson