1. The Fundamental Theorem of Calculus
Lesson 6.2

2. 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

3. 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)?

4. 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

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

6. 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

7. 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)

8. 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

9. Example Consider What is F(x), the antiderivative?
Evaluate F(5) – F(0)

10. Properties Bringing out a constant factor
The integral of a sum is the sum of the integrals

11. Properties Splitting an integral
f must be continuous on interval containing a, b, and c

12. Example
Consider
Which property is being used to find F(x), the antiderivative?
Evaluate F(4) – F(0)

13. Try Another What is Hint … combine to get a single power of x
What is F(x)?
What is F(3) – F(1)?

14. Assignment Lesson 6.2 Page 227
Exercises 1, 5, 9, … 41, 45 (every other odd) Second Day 3, 7, … 47 (every other odd)