1. Fundamental Concepts of Integral Calculus

2. Integral and Derivative Are Complements
Derivative: Give me distance and time, and I’ll give you velocity (speed, rate)
Integral: Give me velocity and time, and I’ll give you distance

3. Indefinite Integral The expression:
read “the indefinite integral of f with respect to x,”
means to find the set of all antiderivatives of f.
x is called the variable of integration
Integrand
Integral sign

4. Table of Indefinite Integrals

5. Rewriting before integrating

6. 0 = -1 + C C = 1 Find the general solution of the equation F’(x) = and
find the particular solution given the point F(1) = 0.
Now plug in (1,0) and solve for C.
0 = -1 + C
Final answer.
C = 1

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

9. 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.

10. Fundamental Theorem of Calculus
If f is a continuous function on the closed interval [a, b] and F is any antiderivative of f, then

11. Indefinite Integrals or Antiderivatives
You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions).