Fundamental Theorems of Integral Calculus

We define integrals as the function of the area bounded by the curve y = f(x), a ≤ x ≤ b, the x-axis, and the ordinates x = a and x =b, where b>a. Let x be a given point in [a,b]. Then 

^b∫_a f(x)dx ∫abf(x)dx represents the area function. This concept of area function leads to the fundamental theorems of integral calculus.

• First Fundamental Theorem of Integral Calculus
• Second Fundamental Theorem of Integral Calculus

First Fundamental Theorem of Integrals

A(x) = b∫af(x)dx∫abf(x)dx for all x ≥ a,

where the function is continuous on [a,b].

Then A'(x) = f(x) for all x ϵ [a,b]

Second Fundamental Theorem of Integrals

If f is continuous function of x defined on the closed interval [a,b] and F be another function such that d/dx

F(x) = f(x) for all x in the domain of f,

then ^b∫_af(x)dx ∫abf(x)dx = f(b) -f(a).

This is known as the definite integral of f over the range [a,b], a being the lower limit and b the upper limit.