Volume can be explained as “how much space does this take up” or “how much water do we need to fill this thing up.” Obviously a good starting reference point for my question here is both the cube and the rectangular prism, whose volume formula is simply length*width*height. Once we get to stuff like cones, pyramids, spheres, and cylinders, formulas start throwing people in for a loop, giving you various hoops that you have to go through. Don’t even get me started on inner tubes ([https://en.wikipedia.org/wiki/Solid_torus](https://en.wikipedia.org/wiki/Solid_torus) ) and vases ([https://en.wikipedia.org/wiki/Solid_of_revolution](https://en.wikipedia.org/wiki/Solid_of_revolution) ). In any case, how did we manage to create formulas for the volumes of solids that aren’t boxes?
In: Mathematics
The first we did was to just measure the volume using one of the methods you said and then taking measurement from various different sizes of the shape we can make a formula which match the measurements. However when calculus were invented it came with a tool we can use to come up with the volume of any mathematical shape. Integration is to find a formula for the area under a graph. When you integrate twice for a three dimentional object you get the volume.
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