# Derivatives and Integrals

by Swetasuria
Tags: derivatives, integrals
 P: 47 What I have learnt in school is that differentiation and integration are opposites. By integrating a function we find the area under the graph. So, integration gives us the area. Differntiation gives slope of the function. If I am right by saying differentiation and integration are opposites, then is area under the graph the opposite of the slope of the graph?
 Sci Advisor HW Helper PF Gold P: 12,016 What dioes "opposite" mean?? Differentiation UNDO what indefinite integration does to a function f(x), that is: Diff(Int(f(x))=f(x), whereas indefinite integration UNDO, up to an arbitrary error constant, what differentiation did to the function f(x). that is: Int(Diff(f(x))=f(x)+some constant.
P: 47
 Quote by arildno Differentiation UNDO what indefinite integration does to a function f(x), that is: Diff(Int(f(x))=f(x), whereas indefinite integration UNDO, up to an arbitrary error constant, what differentiation did to the function f(x). that is: Int(Diff(f(x))=f(x)+some constant.
Pardon me if I sound stupid.
If we undo area under the graph, we get slope of that function then? Now I don't know how to undo the area or if thats even possible. Enlighten me.

HW Helper
PF Gold
P: 12,016
Derivatives and Integrals

 Quote by Swetasuria Pardon me if I sound stupid. If we undo area under the graph, we get slope of that function then?
no.
 Mentor P: 21,251 Swetsuria, Notice that arildno said indefinite integration (no limits of integration). To get the area under the graph of a function, you use a definite integral. The Fundamental Theorem of Calculus state in one of its two parts that differentiation and integration are essentially inverse operations. $$\frac{d}{dx}\int_a^x f(t) dt = f(x)$$ Although the integral in this formula is a definite integral, due to the fact that one of the limits of integration is a variable (x), the integral represents a function of x.
 Quote by Mark44 Swetsuria, Notice that arildno said indefinite integration (no limits of integration). To get the area under the graph of a function, you use a definite integral. The Fundamental Theorem of Calculus state in one of its two parts that differentiation and integration are essentially inverse operations. $$\frac{d}{dx}\int_a^x f(t) dt = f(x)$$ Although the integral in this formula is a definite integral, due to the fact that one of the limits of integration is a variable (x), the integral represents a function of x.