1
$\begingroup$

This question may seem to be off-topic, So before flagging or reporting as off topic, feel free to comment and I will delete this question.

I have recently studied high school calculus and I am still wondering that how are integrals and derivatives related to each other?

Is this just a coincidence?

Or there is a logic that I am still not aware of.

Out of my curiosity I asked this question to my teacher but I didn't get any satisfactory answer.

And one more question,

How definite integrals calculate area of the given function.I asked this too, then my teacher replied that when we integrate $y(x).dx$ ; $dx$ is a small strip of width $dx$, parallel to $y$-axis and $y$ is it's height, so together multiplied it gives area in that rectangular region and the integral sums it up.

But then I found a question on this website asking : Is $\frac {dy}{dx}$ not a fraction? and it turned out that it isn't.Therefore we can't consider $dx$ as width, after all, it's a operator.

So what is going on?

$\endgroup$
1
  • $\begingroup$ The fundamental theorem of calculus simply says that the total change $f(b) - f(a)$ is the sum of all the little changes $f'(x) dx$. When you look at it that way, it seems extremely unsurprising and intuitive. $\endgroup$ Commented Mar 11, 2017 at 11:05

1 Answer 1

Reset to default
3
$\begingroup$

Fundamental Theorem of Calculus:

If$dF(x)/dx = f(x)$, then $\int f(x) dx = F(x) + C$ and $\int_a^b f(x) dx = F(b) - F(a)$

Note (very important and often missed by young students who were not taught to distinguish) -- capital F and lower-case $f$ are different letters and functions.

That says it all -- integration and differentiation are defined as inverse operations.

In practical terms, consider how you do differentiation and integration. For derivatives, you subtract the y values, then divide the delta y by delta x. For integrals and approximations of area, you multiply y values by delta x and then add up the areas. Since division and multiplication are opposite or inverse functions, ans so are subtraction and addition, it seems reasonable that subtract-divide which in the limit leads to the derivative is the inverse of multiply-add which in the limit gives the integral.

No, $dy/dx$ is technically not a fraction. But it is a limit of a fraction; $dy/dx = \lim_{\Delta x \rightarrow 0} \Delta y / \Delta x$.

As you go on in calculus, you will meet techniques of integration where we do substitution/ change of variables and we treat $dy/dx$ algebraically as if it were a fraction. Later in differential equations the terms $dy$ and $dx$ are written independently all the time. This is called abuse of notation but as long as you are reasonably careful it works well and is quite standard.

$\endgroup$

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.