Volumes of revolution, if you didn't know, is basically the volume of the solid you get when you rotate some area of a graph 360 degrees around either the x-axis or y-axis. For example, in the diagram below, when you rotate the shaded bit between the lines x = 0, y = 0, x = 1 and y = 2 about the y-axis (or x-axis for that matter), you get a cylinder. If you don't see how this works, then tape some paper onto a pencil and twirl it around a bit.
Here are the two formulae for working out the volume of the solid formed. The top one is for rotating around the x-axis, the bottom one for the y-axis. In the first equation, a and b are the x-coordinates of the "start" and "end" points of the volume to be calculated, while in the second equation, a and b are the y-coordinates of the "start" and "end" points.
You might be thinking, but wait! How do I integrate with respect to x when I only have ys there, and vice versa? Well, you just have to rearrange the equation. For example, if your equation is y = x^2, and you want to find the volume when you rotate an area under that graph around the x-axis, then you have to rearrange it to get y^2 = (x^2)^2 = x^4, and then substitute x^4 into the first equation. Then all you have to do is solve it like a definite integral.
Now, here's the simple(r) and probably very mathematically incorrect explanation as to why this formula works.
I'm going to put the above graph here again, just so you don't have to scroll back up just to see what I'm blabbing on about.
As you can see, this to-be cylinder either has a height of 2 units and a radius of 1, or vice versa, depending on whether you're going to be rotating the shaded bit about the y-axis or x-axis. For now, we're going to rotate it around the y-axis, because I tend to think of cylinders as being long thin things, because my thinking totally conforms to stereotypes like that. To find the area of this cylinder, then, you could just use our good ol' formula back from year 8 or maybe from primary school if you were lucky enough to have teachers willing to teach that stuff there: the good ol' (pi)(r^2)(h) formula for finding out the volume of cylinders.
Now wait! In the above diagram, since we're rotating about the y-axis, the radius is equal to x. That's how we get the (pi)x^2 part of the formula. Now, as for y, and the whole dy and integral and everything else that's in the volume of integration formula, we have to think outside the orange rectangular box above and start thinking about all kinds of weird irregular shapes.
The (pi)(x^2) part gives the area of any given little slither of area rotated around the y-axis. To actually turn that into volume, we have to multiply that by height as well. We could just multiply by the height of the entire area in the above diagram, but remember: most curves do all kinds of weird things. The radius, or x^2, generally changes as you go along the curve, so you have to multiply by the height of infinitesimally small sections at a time to get the volume. Here's a weird curvy thing that I drew to explain this better:
Note that at y = 2, x, and therefore x^2, is much bigger than it is at x = 1. Every little section along this weird curvy thing has a different x-value too. Therefore, to calculate the total volume of this curve between y = 1 and y = 2, we have to calculate the volume of every tiny little slither of volume between these two points and add them all together.
Thankfully, someone smart invented integration, so we don't have to do all that. I think that the "dy" part of the integration formula is a reference to all the small slithers, a bit like how (delta)y is used to symbolise a small change in y.
Therefore, to recap: the (pi)(x^2) in the integration formula is the circular area of one of the little bits of area under the curve rotated around the y-axis, while the dy is the height of each slither. Then the b and a tell you where to start and stop adding up slithers. And then that fancy integration sign is there because it's fun to draw and makes you feel like a maths genius every time you do so.
For the formula for rotating around the x-axis, it works much the same way: the (pi)(y^2) in the integration formula is the circular area of one of the little bits of area under the curve rotated around the x-axis, while the dx is the width of each slither.
Gosh, I could almost be talking about snakes with the many times that I've said the word "slithers!"
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