0 like 0 dislike
0 like 0 dislike
What the fuck does reversing the derivative operation have to do with summing up infinitely thin(enough) rectangles under a curve?

4 Answers

0 like 0 dislike
0 like 0 dislike
I remember thinking the exact same thing when I learned integration, here's what made it eventually click for me (though this is not a formal proof by any means)

Let the area under the graph of y = f(x) from 0 to x be A(x). If you extend x by a tiny amount h, that's like adding one of those tiny rectangles to the sum - in this case that rectangle will be h \* f(x). When h is small, you expect this to be close to the actual difference of A(x) and A(x + h). In other words:

A(x + h) \~= A(x) + hf(x)

Then by rearranging:

f(x) \~= (A(x + h) - A(x))/h

When h tends to 0, this gives the exact defintion of the derivative, showing that the function of the area under a graph of a function differentiates to give the function itself. A stronger argument would show that the approximation tends to an equation as h tends to 0.
0 like 0 dislike
0 like 0 dislike
If you walk in a straight line at various speeds, the distance you travel can be determined by a weird kind of sum of how fast you were walking and for how long you were walking that fast. Does that make sense? If you walk 5mph for an hour you've walked 5 miles.

That is intuitively equivalent to this idea. You've summed up "rates of change in position" i.e. "the derivative of position" i.e. "how fast I'm walking along the straight line" according to how long you walked that fast, and you got a distance.

"The sum of small changes times how long we were changing that fast tells us how far we've moved" rather "how far we've moved is the sum of small changes times how long we were changing that fast"- this is the fundamental theorem of calculus.

Edit:

This is also the intuition behind the Stokes theorem, it's special case Green's Theorem, and also the generalized Stokes theorem (of which the Fundamental theorem of calculus is actually just a special case).
0 like 0 dislike
0 like 0 dislike
Suppose you didn't know how much money you had in your bank account at the moment, but you did know how much you had a month ago, and you knew what you added and subtract each day since then?

If you added up all those changes over time, then you could add that to what you had a month ago and you would know what you have now.
0 like 0 dislike
0 like 0 dislike
The ELI5 answer is:

If you multiply the slope by a tiny x-slice you get a small change in y value. If you multiply a y value by a tiny x-slice you get a small change in area.

Integrate y-values, get an area.

Integrate slopes, get a y-value.

Related questions

0 like 0 dislike
0 like 0 dislike
2 answers
thebandapp asked Jun 21
If you subscribe to a service that is £100 a year for a 5 year term, and you also get £50 sign up discount and one year free, which of the following do you pay?
thebandapp asked Jun 21
0 like 0 dislike
0 like 0 dislike
4 answers
0 like 0 dislike
0 like 0 dislike
27 answers
DrRamAmbatkar asked Jun 21
How do people come up with mathematical arguments on the fly?
DrRamAmbatkar asked Jun 21
0 like 0 dislike
0 like 0 dislike
3 answers
FlashoverRec asked Jun 21
EXTREME novice here. How do people (mathematicians, physicists, engineers, etc.) come up with their own formulas?
FlashoverRec asked Jun 21
0 like 0 dislike
0 like 0 dislike
10 answers
Mark_Weinberger asked Jun 21
How do you not get discouraged after messing up math assessments?
Mark_Weinberger asked Jun 21

33.4k questions

135k answers

0 comments

33.7k users

OhhAskMe is a math solving hub where high school and university students ask and answer loads of math questions, discuss the latest in math, and share their knowledge. It’s 100% free!