# How can you prove the integral without knowing the derivative?

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In summary, the conversation discusses the relationship between differentiation and integration, and whether it is possible to compute an integral without knowing the derivative or the fundamental theorem of calculus. Examples and explanations are given to showcase the connection between the two concepts, and a detailed calculation is shown for computing a definite integral using "first-principle" without utilizing the fundamental theorem of calculus.
Trying2Learn
TL;DR Summary
How can you prove the integral without knowing the derivative
Hello

(A continued best wishes to all, in these challenging times and a repeated 'thank you' for this site.)

OK, I have read that Newton figured out that differentiation and integration are opposites of each other.
(This is not the core of my question, so if that is wrong, please let it go.)

I can work out the derivative of, say, f(x) = x-squared.

However, I am having trouble proving the integral of, say, x-cubed.

If it is true, that it was 'discovered' that differentiation is anti-integration or integration is anti-derivative...

If that is true that it was 'discovered' and that people were studying differentiation and integration SEPARATELY, then I should be
able to derive the integral of say, x-cubed.

But I cannot do it without the first fundamental theorem of the calculus (which, assumes already that the two are opposites).

First, is this a ridiculous question?

If not, could someone work out one example? Something simple like the indefinite integral of say: x-cubed divided by 3

Thank you

You can look at the methods for example. A derivative is the quotient ##\dfrac{f(p+h)-f(p)}{h} \longrightarrow \dfrac{df}{dx}## where ##h \longrightarrow dx## is getting smaller and smaller, and ##f(p+h)-f(p) \longrightarrow df##, which is the opposite of Riemannian sums: ##\displaystyle{\sum f(x_h) \cdot h}## where the interval lengths ##h \longrightarrow dx## again. One is a quotient and the other a multiplication.

You can also have a look at easy examples like ##f(x)=c,\; f(x)=x ## or ##f(x)=x^2## and observe the connection of both, and then think about a generalisation.

Trying2Learn said:
Summary::
If not, could someone work out one example? Something simple like the indefinite integral of say: x-cubed divided by 3

What do you mean by "indefinite integral" if not just anti-derivative?

What you can do without derivatives, though is compute the definite integral ##\int_a^b x^3 dx=\frac{1}{4}(b^4-a^4)## from the Riemann sum definition of an integral.

I understand what both are saying.
(And this is where it may be that the question is silly.)

But I can take the difference of the ratio of the function at two points divided by the difference in the points and get an approximation to the derivative.

I can actually show the derivative of f(x) = x-squared, is 2x.

But can I show the reverse WITHOUT knowing the anti-derivative, WITHOUT knowing the derivative is the opposite of the Riemann sum.

So why can't I formulate the integral WITHOUT knowing about the derivative?

I am confused.

You can calculate ##\displaystyle{\int_a^b} x^2\,dx## without knowing the derivative of ##f(x)=x^2## simply by Riemann sums, or without knowing that ##\dfrac{d}{dx}\dfrac{1}{3}x^3 =x^2.##

Here is an example of how you would go about computing a definite integral of ##f(x) = x^2## from "first-principle" without utilizing the fundamental theorem of calculus. I will only show the calculations for the right evaluated Riemann sum, as the calculations for the left evaluated Riemann sum is identical to it.
\begin{align*} A(x) &= \int_0^xf(s)\,ds \\ &:= \lim_{N\rightarrow\infty}\frac{x-0}{N}\sum_{k=1}^Nf\Big(k\frac{x-0}{N}\Big) \\ &= \lim_{N\rightarrow\infty}\frac{x}{N}\sum_{k=1}^N\Big(k\frac{x}{N}\Big)^2 \\ &= x^3\lim_{N\rightarrow\infty}\frac{1}{N^3}\sum_{k=1}^Nk^2 \\ &= x^3\lim_{N\rightarrow\infty}\frac{1}{N^3}\frac{N(N+1)(2N+1)}{6} \\ &= x^3\lim_{N\rightarrow\infty}\Big(\frac{1}{3} + \frac{1}{2}\frac{1}{N} + \frac{1}{6}\frac{1}{N^2}\Big) \\ &= \frac{1}{3}x^3.\end{align*}
Where third last equality is valid by Faulhaber's Formula <LINK>, particularly the Faulhaber formula for
$$\sum_{k=1}^Nk^2 = \frac{N(N+1)(2N+1)}{6}.$$

weirdoguy, etotheipi, Stephen Tashi and 2 others
Thank you very much, everyone!

## 1. How can you prove the integral without knowing the derivative?

The fundamental theorem of calculus states that the integral of a function can be found by evaluating its antiderivative at the upper and lower limits of integration. So, even if we do not know the derivative of a function, we can still find its integral using this theorem.

## 2. Can you give an example of proving an integral without knowing the derivative?

Yes, for example, if we want to find the integral of the function f(x) = x^2 from x = 0 to x = 2, we can use the fundamental theorem of calculus. The antiderivative of x^2 is x^3/3, so the integral is (2^3/3) - (0^3/3) = 8/3.

## 3. Is there any other method to prove an integral without knowing the derivative?

Yes, there are other techniques such as substitution, integration by parts, and partial fractions that can be used to evaluate integrals without knowing the derivative. However, these methods may not always be applicable and may require advanced mathematical knowledge.

## 4. What are some limitations of proving an integral without knowing the derivative?

One major limitation is that the fundamental theorem of calculus only applies to continuous functions. If the function is not continuous, then this method cannot be used to prove the integral. Additionally, some integrals may be very difficult to evaluate without knowing the derivative, making it challenging to use this approach.

## 5. How does proving an integral without knowing the derivative relate to real-life applications?

In real-life applications, we often come across situations where we need to find the area under a curve or the total amount of something. In these cases, we can use integrals to solve the problem, and the fundamental theorem of calculus allows us to do so even if we do not know the derivative of the function. This makes it a useful tool in many fields such as physics, engineering, and economics.

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