I thought this was pretty neat:

The source for the text above is by anonymous_28657 from Reddit, who heard it from a Hungarian colleague, who heard it from someone else... It also shows up as a way to annoy your Physics prof (see #14), and the technique is described in this book. If you can afford it, you can get yourself a planimeter to help with your definite integrals.

The source for the text above is by anonymous_28657 from Reddit, who heard it from a Hungarian colleague, who heard it from someone else... It also shows up as a way to annoy your Physics prof (see #14), and the technique is described in this book. If you can afford it, you can get yourself a planimeter to help with your definite integrals.

Awesome, made me laugh out loud.

this is not so strange. i know a professor of numeric mathematics that by a similar method calculated the Area of a vertebra, using thin foils and than ploting the shape of the foil on a 80g/m^2 paper and weighting the paper

Does that mean there is actually a way to relate weight to area?

in europe we measure paper thickens in g/m^2 (weight/area, grams/meters^2), now do the math XD

If you don't trust the given weight you can just cut out a piece of the paper you used with a known area and weigh that.

We totally did this in my analytic chem lab less on 1996! you cut out (accurately) a rectangle of paper weigh it, then you cut out the shape of the curve and weigh that. Create a proportion and you have your integral value.

Of course now we have computers that will use a fancy pants algorithm, but that's what you had to do when you were dealing with equipment that gave analog outputs.

wow. epic

Yes, this is how medical researchers did approximate integration before M.M. Tai invented the trapezoid rule in 1994 (google it)

My high school physics teacher told us about doing this for his Masters back in the day. For curves that they didn't have an equation for they would draw them precisely on really big paper, weigh the paper and then cut out the part above the graph and weigh it again to get the area under the curve.

Thought it was cool in a "oh how crazy things were back in the day" kind of way.

I suspect this is still similar to the technique for calculating the center of area for continents, nations, states, and such.

I always wonder if they take into account the projection of the map, or the curvature of the planet, especially on larger areas.

As a petroleum engineering student, we used this method to calculate the volume of oil reservoirs given the contour maps. We would cut out the shapes for the different contours and find the mass of each contour and then proportionately get the area. Multiplying by each contour's thickness and summing for all contours gave the reservoir's volume.

My boss, a long standing chemist says that this is the technique they really used to use ...

paper is relatively uniform, and due to inaccuracy of measuring the points with a ruler this really did work.

I thought he was joking with me, but apparently, according to him, this really was the best way ... :)

When all you have is an analytical balance, everything looks like a mass problem.

I have had students do this several times. Even today, it is one of the most efficient methods for determining the area under a curve (the definite integral), an important value for many kinds of spectroscopy.

This is a classic!!!

Wow, I'm surprised to see this here. This is a story that nobody believes me when I'm telling it. But it's all true, I was there, I saw it happening many moons ago. :D

I had a chemistry teacher who used a triple-beam balance to determine the number of Scantrons he had instead of counting them out by hand. He had 150 students and knew that 150 Scantrons had a certain mass.

One of my physics profs told us that the weight depends on how much you've touched the paper (yes, that was a reference to greasy nerds), so if you want it to work you have to use gloves, ideally in a cleanroom ;)

Or sculpt the plot, and measure the volume with water displacement

Of course, paper cut-outs, 3d ball-and-springs (with realistic length and force-constant scales), and casual conversation with the crystallographers is how Watson and Crick did it. I have a prof basing his grad-level classical mechanics textbook on teaching geometric visualization for the first two chapters.