1. The Fundamental Theorems of Calculus
Lesson 5.4

4. Don't let this happen to you!

5. Average Value Theorem Consider function f(x) on an interval Area =
a b
Consider function f(x) on an interval
Area =
Consider the existence of a value for x = c such that a < x < b and c

6. Average Value of Function
a b
Also The average value is c

7. Area Under a Curve
f(x) a b
We know the area under the curve on the interval [a, b] is given by the formula above
Consider the existence of an Area Function

8. Area Under a Curve
f(x)
A(x) a x b
The area function is A(x) … the area under the curve on the interval [a, x]
a ≤ x ≤ b
What is A(a)?
What is A(b)?

9. The Area Function If the area is increased by h units
f(x)
A(x+h)
x x+h a b h
If the area is increased by h units
New area is A(x + h)
Area of the new slice is A(x + h) – A(x)
It's height is some average value ŷ
It's width is h

10. The Area Function The area of the slice is
Now divide by h … then take the limit
Divide both sides by h
Why?

11. The Area Function
The left side of our equation is the derivative of A(x) !!
The derivative of the Area function, A' is the same as the original function, f

12. Fundamental Theorem of Calculus
We know the antiderivative of f(x) is F(x) + C
Thus A(x) = F(x) + C
Since A(a) = 0
Then for x = a, 0 = F(a) + C or C = -F(a)
And A(x) = F(x) – F(a)
Now if x = b
A(b) = F(b) – F(a)

13. Fundamental Theorem of Calculus
We have said that
Thus we conclude
The area under the curve is equal to the difference of the two antiderivatives
Often written

14. First Fundamental Theorem of Calculus
Given f is
continuous on interval [a, b]
F is any function that satisfies F’(x) = f(x)
Then

15. First Fundamental Theorem of Calculus
The definite integral can be computed by
finding an antiderivative F on interval [a,b]
evaluating at limits a and b and subtracting
Try

16. Area Under a Curve
Consider
Area =

17. Area Under a Curve
Find the area under the following function on the interval [1, 4]

18. Second Fundamental Theorem of Calculus
Often useful to think of the following form
We can consider this to be a function in terms of x
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19. Second Fundamental Theorem of Calculus
This is a function
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20. Second Fundamental Theorem of Calculus
Suppose we are given G(x)
What is G’(x)?

21. Second Fundamental Theorem of Calculus
Note that
Then
What about ?
Since this is a constant …

22. Second Fundamental Theorem of Calculus
Try this

23. Second Fundamental Theorem of Calculus
An application from Differential Equations
Consider
Now integrate both sides
And

24. Assignment
Lesson 5.4
Page 327
Exercises 1 – 49 odd