# Integration and differentiation

1. Mar 18, 2012

### jd12345

Why is antiderivative and area under the curve the same thing? Its not at all intuitive to me

Derivative is the slope at a point and its opposite is area?? Can someone just explain me why when we are finding an antiderivative, we are actually finding area under the curve

i dont buy the fact that slope of a curve and area under the curve ae opposite of each other

2. Mar 18, 2012

### Alpha Floor

Oh my! Right now you're undergoing your very first mathematical crisis! I've been there some years ago and trust me, eventually you will understand it.

I recommend you read a book by William Dunham, "The mathematical universe" chapters D, K and L.

It's something you must discover by your own. For now I can just say that you are confusing "antiderivative" with "integral", which are quite different things. The first is a function, while the later is a real number.

3. Mar 18, 2012

### HallsofIvy

Staff Emeritus
The standard proof is this- look at the graph of y= f(x), a continuous. Let the area under the curve, above y= 0, and between x= a and x, be F(x). Let $x^*$ be a value x at which F takes its maximum, $x_*$ a value at which F takes its minimum on [a, x]. Then we must have $f(x_*)(x- a)\le F(x)\le f(x^*)(x- a)$. Then
$$\frac{F(x_*)}{x- a}\le f(x)\le\frac{F(x^*)}{x- a}$$
Because both x* and $x_*$ are between a and x, if we take the limit as x goes to a, x* and $x_*$ will also both go to a. But then,
$$\lim_{x\to a}\frac{F(x_*)}{x-a}= \lim_{x\to a}\frac{F(x^*)}{x- a}= \frac{dF}{dx}$$ and we have
$$\frac{dF}{dx}\le f(x)\le \frac{dF}{dx}$$

Last edited by a moderator: Mar 18, 2012