Does anyone know how to find the area & volume of disks and washers?!!!

Question

kainui · Answer

So first off lets look at the circumference and area formulas for a circle. $$C=2\pi r$$$$A=\pi r^2$$These themselves are related to each other by integration and differentiation! The derivative of area is circumference and the integral of circumference is the area! Why? Let's look at it from an integration standpoint. Imagine you have a circle with a circumference that you want the area of. If you don't know the area formula, how can we get there? Well, as it turns out it's pretty simple. If you just add up the circumferences of every circle inside that circle, all those circumferences will add up to the area. That means an infinite number of circles though, so we end up with an infinitely small difference in radius between each circle, which we call dr. So when we multiply circumference by dr we get the area of a circle with an infinitely small radius. This by itself isn't very useful, but when we add the integral sign, "S" which just means sum, we add up all the circumferences or all circles between our two ends. So if you are to integrate from 0 to R (R can mean whatever radius) you'll get the area of a circle with radius R. If you integrate from some number higher than zero to another, you'll end up with a washer since you're only adding up the circumferences of circles between those two radii. Similarly, you can just use the area formula for a circle and subtract the area of a smaller circle to make a washer. $$\int\limits_{}^{}C=\int\limits_{0}^{R}2\pi dr$$ Pretty straightforward so far? So hopefully that helps in explaining the disk and washer methods! http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx I think these pictures can explain it better than I can, but what you don't understand I'll be happy to explain more in depth!