i do get that Riemann sums is sum of infinite triangles with with infinitely small length. But definite integral is completely different you are taking anti derivative of f(x) at b and subtract anti derivative of f(x) at point a.
Explanation one that fails to make sense is this: definite integral is just notation it's same thing as limit of Riemann sums. And i was fooled till all of the sudden and quietly my class started using antiderivatives..symbol were same so it took a week for me confront my confusion.
Explanation two is this answer…but it fails to explain or just assume either that F'(x) = f(x) Or… if proving later then it assumes definite integral equals area. Basically doesn't prove one without not relying other which happen to be also be in need of proof or explanation to me.
So in short can you explain/proof how or why:
- Definite Integral ( the difference of antiderivatives ) equals area F(x)=A(X)
- F'(x)=f(x)
if you can show me proof of one without relying on other, i.e. F'(x) = f(x), without relying of F(x) = A(x) then it will be enough.
I have been trying to understand this for 40+ days..tutor, professor, 3 textbooks, and google all failed me.
Best Answer
Given a continuous function $f:\>[a,b]\to{\mathbb R}$ we can define its integral over $[a,b]$ in various ways as a limit of Riemann sums. Denoting this limit by $$\int_{[a,b]} f(x)\>{\rm d}x$$ we have $$\int_{[a,b]} f(x)\>{\rm d}x=\lim_{\ldots}\>\sum_{k=1}^N f(\xi_k)(x_k-x_{k-1})\ ,$$ where I don't bother now about the exact definition. At any rate, when $f(x)\geq0$ on $[a,b]$ we can interpret this integral "intuitively" as "area under the curve $y=f(x)$".
Of course we want to compute this integral when $f(x)$ is some given expression in the variable $x$. In order to obtain a general procedure Newton had the following fundamental idea: We consider momentarily the upper limit $b$ as variable, i.e., we are looking at the area function $$A(u):=\int_{[a,u]} f(x)\>{\rm d}x\ .$$ A more or less geometric limit argument then shows that this area function has the property $$A'(u)=f(u)$$ for all $u$ in the range at hand. This means that $A(\cdot)$ is an antiderivative of $f$, and it is then not difficult to show that $$\int_{[a,b]} f(x)\>{\rm d}x=F(b)-F(a)\ ,\tag{1}$$ where $F$ can be any antiderivative, or: primitive, of $f$ on $[a,b]$.
The difference on the right hand side of $(1)$ does only depend on the given $f$, and not on the particular primitive $F$ chosen. We therefore want a notation for this difference that exhibits only $f$, as well as $a$ and $b$. The established notation is, as we all know, $$\int_a^b f(x)\>dx\ .\tag{2}$$ The expression $(2)$ has the following meaning: Compute, by whichever means, a primitive $F(x)$ of the function term $f(x)$, and return the difference $F(b)-F(a)$.
In this way the fundamental theorem of calculus assumes the (only seemingly tautological) form $$\int_{[a,b]} f(x)\>{\rm d}x=\int_a^b f(x)\>dx\ .$$ On the left hand side we have a limit of Riemann sums, and on the right hand side a difference $F(b)-F(a)$ for some primitive of $f$.