Theorem 1.
Let f be a continuous function of the closed interval [a, b] and let A(x) be the area function. Then, A'(x) f(x) for all x ∈ [a1 b]
Second fundamental theorem of integral calculus
Theorem 2.
Letf be continuous function defined on the closed interval [a, b] and F be an antiderivative off.
Then, \(\int^x_a\) f(x) dx = [F(x)]_{a}^{b} = F(b) - F(a)
In other words \(\int^x_a\) f(x) dx = (Value of the anti-derivative F off at the upper limit b) - (Value of the same anti-derivative at the lower limit a)
Remarks:
(1) Second fundamental theorem of integral calculus is very useful, because it gives us a method for calculating the definite integral more easily.
(2) In \(\int^x_a\) f(x) dx, the function f needs to be well defined and continuous in [a, bi.
