American version from below:

The "you learn limits in like, 9th grade" comment reminds me of this one:

Two mathematicians are in a bar. The first one says to the second that the average person knows very little about basic mathematics. The second one disagrees, and claims that most people can cope with a reasonable amount of math. The first mathematician goes off to the washroom, and in his absence the second calls over the waitress. He tells her that in a few minutes, after his friend has returned, he will call her over and ask her a question. All she has to do is answer one third x cubed.

She repeats "one thir -- dex cue"?

He repeats "one third x cubed".

She says, "one thir dex cuebd"?

Yes, that's right, he says. So she agrees, and goes off mumbling to herself, "one thir dex cuebd...".

The first guy returns and the second proposes a bet to prove his point, that most people do know something about basic math. He says he will ask the blonde waitress an integral, and the first laughingly agrees. The second man calls over the waitress and asks "what is the integral of x squared?".

The waitress says "one third x cubed" and while walking away, turns back and says over her shoulder "plus a constant!"

I've heard a variation of that were the professor is at a bar with a couple of visiting professors. He tells them, in this town pretty much everyone is smart and he proves it by asking the waitress. (The rest of the joke is the same.)

I think the integral is

        4 * integrate sqrt(r^2 - x^2) w.r.t. x from x=0 to x=r

which is from the equation of circle with radius r centered at the origin:

        x^2 + y^2 = r^2
        => y = sqrt(r^2 - x^2)

any other way achieve the same using integration?

You can arrive at the same by slicing in nearly any manner you wish: vertical dx strips, horizontal dy strips, radial dtheta strips, concentric rings treated as rectangles, diagonal strips, literally anything you wish.

If you've never set up and worked a few, try the dx version and the dy version, then maybe fiddle with a few others. They all work.

The concentric rings one is trivial to work out. Integrate over r from 0 to R. Each ring at radius r has thickness dr, and has length 2 pi r, treated simply as a rectangle, which is "close enough" since each is arbitrarily thin. Then the integral is

int_0^R 2 pi r dr = 2pi r^2/2 from 0 to R = pi R^2.

A really pretty one cuts the circle into wedges, then rearranges them by alternating direction to make a "rectangle" approx r high, approx pi * r wide, with bumpy top and bottom. In the limit this has area pi * r * r, and can be shown to kids without needing calculus.

And it's not magic or circular, since pi is defined (in this case...) as the ratio of circumference to diameter.

\int_0^2π{(1/2)r^2dθ}

Value inside the integral is area of thin triangle of height r and base rdθ.

I don't quite get the punchline. I understand how if you integrate in the wrong direction you get a negative area, but is there a double meaning here?

I believe the implication is that all the plumbers are advanced degree holders, not just the engineer.

-The idea is that all the plumbers are academics looking for the easy life. :)

The limits are the shackles of oppression, the integral is the proletariat and switching is disavowing capitalism and seizing the means of production.

That was completely unhelpful but also hilarious.

As such, a perfect russian joke.

Thank you, that's what I was going for!

I shall never look at calculus the same way again. :) Thank you, comrade!

Brilliant !

Too many levels of indirection.

From the top:

https://news.ycombinator.com/item?id=25866357

They're all former academics/mathematicians!

They are all engineers in disguise.

I think I had already heard that one from a math teacher

Caught me off-guard :D