1. Fundamental Theorem of Calculus Indefinite Integrals
Fundamental Theorem of Calculus
Indefinite Integrals

3. Example: Evaluate A(x)
Area between the graph of f(x) and the x-axis over the interval [2,x]

4. Using geometry:

5. Using integration:

6. Fundamental Theorem of Calculus

7. Fundamental Theorem of Calculus – part 1:
Fundamental Theorem of Calculus – simplest form:
Fundamental Theorem of Calculus – more general form:

8. Fundamental Theorem of Calculus - part 2:
Suppose that f is bounded on the interval [a,b], and that F is an antiderivative of f, i.e.,  F’ = f.     Then:

9. Example 1:
Solution:

10. Example 2:
Solution:

11. Example 3:
Solution:

12. More practice problems with solutions:

13. 4.5
Substitution Rule

16. Example 1: Find  x3 cos(x4 + 2) dx. Solution:
We make the substitution u = x4 + 2 because its differential is du = 4x3 dx, which, apart from the constant factor 4, occurs in the integral.
Thus, using x3 dx = du and the Substitution Rule, we have
 x3 cos(x4 + 2) dx =  cos u  du
=  cos u du

17. Example 1 – Solution cont’d = sin u + C = sin(x4 + 2) + C
Notice that at the final stage we had to return to the original variable x.

19. Example 2: Evaluate . Solution:
Let u = 2x Then du = 2 dx, so dx = du.
So: 4