How does the fundamental theorem of calculus connect derivatives and integrals?

Question

Cameron Alexander · Accepted Answer

The fundamental theorem of calculus shows that differentiation and integration are reverse processes of each other.   Let us look at the statements of the theorem.
(I) #d/dx int_a^x f(t)dx=f(x)#
(II) #int f'(x)dx=f(x)+C# As you can see above, (I) shows that integration can be undone by differentiation, and (II) shows that differentiation can be undone by integration (with a loss of the information of C).

Scarlett Allison · Answer

undefined The Fundamental Theorem of Calculus connects derivatives and integrals by stating that if a function $ f(x) $ is continuous on a closed interval $[a, b]$ and $ F(x) $ is an antiderivative of $ f(x) $ on that interval, then:

1. The definite integral of $ f(x) $ from $ a $ to $ b $ is equal to $ F(b) - F(a) $.
$$ \int_a^b f(x) \, dx = F(b) - F(a) $$

2. If $ F(x) $ is any antiderivative of $ f(x) $, then the indefinite integral of $ f(x) $ is given by $ F(x) + C $, where $ C $ is a constant.
$$ \int f(x) \, dx = F(x) + C $$

In simpler terms, the Fundamental Theorem of Calculus establishes a connection between the process of differentiation and integration, showing that integration (finding the area under a curve) is essentially the reverse process of differentiation (finding the slope of a curve). It allows us to evaluate definite integrals by finding antiderivatives of functions and applying them to the limits of integration.