1. Aim: What is the Fundamental Theorem of Calculus?
Do Now:
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2. Connection: Differentiation & Integration
Two branches of calculus:
Differentiation - rate of change
Integration – accretion (area)
Inverses of each other Δy Δx
secant Δy Δx
area of rectangle
tangent
area of region
precalculus
precalculus

3. Connection: Differentiation & Integration
Calculus is the study of limits
derivative of a function
two most important limits
definite integral
Fundamental Theorem of Arithmetic
whole numbers can be factored into product of primes
Fundamental Theorem of Algebra
nth degree polynomial has n roots

4. The Fundamental Theorem of Calculus
If a function of f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b], then
Guidelines
You now have a way to evaluate a definite integral without using the limit of a sum.
Use the following notation
It is not necessary to use the constant of integration C

5. Evaluating a Definite Integral

6. Evaluating an Absolute Value

7. Finding Area of Region
Find the area of the region bounded by the graph of y = 2x2 – 3x + 2, the x-axis, and the vertical lines x = 0 and x = 2.
Integrate [0, 2]
Find F(x)
Evaluate F(2) – F(0)

8. Mean Value Theorem of Integrals
If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that c
f(c) f
= average value of f on [a, b] a b
somewhere between the inscribed and circumscribed rectangles there is a rectangle whose area is equal to the area of the region under the curve.

9. Average Value of a Function
If f is integrable on the closed interval [a, b], then the average value of f on the interval is
Find the average value of f(x) = 3x2 – 2x on the interval [1, 4]

10. Average Value of a Function
Find the average value of f(x) = 3x2 – 2x on the interval [1, 4]
3x2 – 2x = 16
c = 8/3 c
f(c)
average value

11. Model Problem
At different altitudes in earth’s atmosphere, sound travels at different speeds. The speed of sound s(x) (in meters per second) can be modeled by
where x is the altitude in kilometers. What is the average speed of sound over the interval [0, 80]?

12. Model Problem
Sum of 5 integrals -

13. Accumulation Function
Definite Integral as a Number
Definite Integral as a Function of x
Constant
F is a function of x
new variable of integration
f is a function of x
f is a function of t
Constant
Constant
Accumulation function: area accumulates under a curve from fixed value of (t = a) to a variable value (t = x)
Note: definite integral is not function of variable of integration

14. Model Problem
Evaluate
option 1
option 2

15. 2nd Fundamental Theorem of Calculus
If f is continuous on an open interval I containing a, then, for every x in the interval,
Evaluate
Caution: if the upper limit is a function of x, ex. x2, then the answer is multiplied by the derivative of the upper limit term.

16. Model Problem
Find the derivative of F(x)
chain rule
verification

17. Model Problem

18. Model Problem

19. Model Problem

20. Model Problem

21. Model Problem

22. Model Problem

23. Model Problem

24. Model Problem