1. 6.2 Definite Integrals

2. When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
subinterval
partition
Subintervals do not all have to be the same size.

3. subinterval
partition
If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by
As gets smaller, the approximation for the area gets better.
if P is a partition
of the interval
Definition – The Definite Integral as a Limit of Riemann Sums

4. is called the definite integral of
over
If we use subintervals of equal length, then the length of a subinterval is:
The definite integral is then given by:

5. Leibnitz introduced a simpler notation for the definite integral:
Note that the very small change in x becomes dx.

6. variable of integration
upper limit of integration
Integration
Symbol
integrand
variable of integration
(dummy variable)
lower limit of integration
It is called a dummy variable because the answer does not depend on the variable chosen.

7. Examples: Express the limit as a definite integral.2 1

8. So, if f(x) is nonnegative and integrable over [a, b], then
the area under the curve y = f(x) from a to b is:

9. Area
We have the notation for integration, but we still need to learn how to evaluate the integral.

10. Since rate . time = distance:
In section 5.1, we considered an object moving at a constant rate of 3 ft/sec.
Since rate . time = distance:
If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.
Time
Velocity
After 4 seconds, the object has gone 12 feet.

11. Theorem 2 – The Integral of a Constant, p. 280
Time
Velocity
Theorem 2 – The Integral of a Constant, p. 280
If c is a constant function on [a, b], then

12. If the velocity varies:
Distance:
(Constant = 0 since s = 0 at t = 0)
After 4 seconds:
The distance is still equal to the area under the curve!
Notice that the area is a trapezoid.

13. What if:
We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example.
It seems reasonable that the distance will equal the area under the curve.

14. The area under the curve
We can use anti-derivatives to find the area under a curve!

15. Example:
Find the area under the curve from x=1 to x=2.
Area under the curve from x=1 to x=2.
Area from x=0
to x=2
Area from x=0
to x=1

16. Example:
Find the area under the curve from x=1 to x=2.
To do the same problem on the TI-84:
Y =
Press
2ND
Press
TRACE
Then Press
(CALC)
Type: X2
Set
Lower Limit = 1
Upper Limit = 2
Graph
Press

18. Find the area between the x-axis and the curve from to .
Example:
Find the area between the
x-axis and the curve
from to
pos.
neg.
On the TI-84:
If you use the absolute value function, you don’t need to find the roots.

19. In other words, when f(x) ≤ 0
If an integrable function is non-positive, then the area below the x-axis is the absolute value of that area.
pos.
neg.
In other words, when f(x) ≤ 0
So, for any integrable function,

20. Example: Use the graph of the integrand and areas to evaluate the integral.
First, graph the function.
Notice the radius of the
semicircle is 3. p

21. Now, let’s look at area another way:
Let area under the curve from a to x.
(“a” is a constant)
Then:

22. max f
The area of a rectangle drawn under the curve would be less than the actual area under the curve.
min f
The area of a rectangle drawn above the curve would be more than the actual area under the curve. h

23. As h gets smaller, min f and max f get closer together.
This is the definition of derivative!
initial value
Take the anti-derivative of both sides to find an explicit formula for area.

24. p As h gets smaller, min f and max f get closer together.
Area under curve from a to x = antiderivative at x minus antiderivative at a. p

25. p