integration and differentiation

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

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.

HallsofIvy

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 [itex]x^*[/itex] be a value x at which F takes its maximum, [itex]x_*[/itex] a value at which F takes its minimum on [a, x]. Then we must have [itex]f(x_*)(x- a)\le F(x)\le f(x^*)(x- a)[/itex]. Then
[tex]\frac{F(x_*)}{x- a}\le f(x)\le\frac{F(x^*)}{x- a}[/tex]
Because both x* and [itex]x_*[/itex] are between a and x, if we take the limit as x goes to a, x* and [itex]x_*[/itex] will also both go to a. But then,
[tex]\lim_{x\to a}\frac{F(x_*)}{x-a}= \lim_{x\to a}\frac{F(x^*)}{x- a}= \frac{dF}{dx}[/tex] and we have
[tex]\frac{dF}{dx}\le f(x)\le \frac{dF}{dx}[/tex]