The other part of the Fundamental
Theorem of Calculus (FTC 1)
also relates differentiation and integration, in a slightly
different way.
Fundamental
Theorem of Calculus (Part 1)
If f is a continuous function on [a,b], then the
integral function g defined by g(x)=∫xaf(s)ds
is continuous on [a,b], differentiable on
(a,b), and g′(x)=f(x).
What we will use most from FTC 1
is that ddx∫xaf(t)dt=f(x).
This says that the derivative of the integral
(function) gives the integrand; i.e. differentiation
and integration are inverse operations, they cancel
each other out. The integral function is an anti-derivative.
In this video, we look at several examples using FTC 1.
This video will show you why FTC 1
makes sense.
Notice: The notation ∫f(x)dx, without any upper and lower
limits on the integral sign, is used to mean an anti-derivative of
f(x), and is called the indefinite
integral. This means that
∫cos(x)dx=sin(x)+c, and we don't have to use the capital
F any longer.