## Derivatives and Integrals

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?
 PhysOrg.com science news on PhysOrg.com >> 'Whodunnit' of Irish potato famine solved>> The mammoth's lament: Study shows how cosmic impact sparked devastating climate change>> Curiosity Mars rover drills second rock target
 Recognitions: Gold Member Homework Help Science Advisor 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.

 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.

Recognitions:
Gold Member
Homework Help

## 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 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.