Fundamental Concepts of Integral Calculus

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

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

Table of Indefinite Integrals

Rewriting before integrating

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

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

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.

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

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).