This week in AP Calc we discussed solids of revolution. We used our knowledge of integrals to find the volume of a function that can be formed after a solid is rotated about a certain axis. The disc method was pretty easy to grasp, but the washer method was a little bit more complex and required much more thought. I figured out that drawing the actual rectangle and how it would physically rotate about whichever line was indicated really helped me to understand exactly what I was doing.
When using both the disc and washer methods it is always important to remember that you are going to be forming cylinders for the most part because you are rotating a rectangle (rectangle method of finding the area under the curve) about a certain axis. The height of the cylinder to be formed will be the change in x or dx while the radius will be the curve and function itself. Basically, if you sketch the graph and close the region in which you are told the solid is bound, you can also draw in a rectangle to represent the solid that will be rotated. So, the width of the rectangle when it's rotated about the x-axis will be dx while the height of the rectangle will be f(x). The opposite is true when the solid is being rotated around the y-axis. It is also very important to remember the volume equation for a cylinder because that is where the differential equation comes from. Volume of a Cylinder = (pi)r^2h. Therefore the differential equations for the disc method on both the x-axis and y-axis are:
When using both the disc and washer methods it is always important to remember that you are going to be forming cylinders for the most part because you are rotating a rectangle (rectangle method of finding the area under the curve) about a certain axis. The height of the cylinder to be formed will be the change in x or dx while the radius will be the curve and function itself. Basically, if you sketch the graph and close the region in which you are told the solid is bound, you can also draw in a rectangle to represent the solid that will be rotated. So, the width of the rectangle when it's rotated about the x-axis will be dx while the height of the rectangle will be f(x). The opposite is true when the solid is being rotated around the y-axis. It is also very important to remember the volume equation for a cylinder because that is where the differential equation comes from. Volume of a Cylinder = (pi)r^2h. Therefore the differential equations for the disc method on both the x-axis and y-axis are:
For example, you are given a problem asking to find the volume of the solid when it's rotated about the x-axis and is bound by y = 4-x^2 and y = 0 and you are to find this from [0,2].
We know that f(x) = 4-x^2, 0.
We also know that 4-x^2 is the top most function while 0 is the bottom function, so we would take [4-x^2] - [0] and get 4-x^2 for the final f(x) function. Now, we plug and chug.
pi * PRETEND THERE IS AN INTEGRAL SIGN HERE [(4-x^2)^2] dx ---> 256pi/15
The washer method, like I said earlier, is a little more complex because you have to take into consideration the fact that part of the solid will be hollowed out. Essentially, you take the same equation for the disc method because it is still based off of the volume formula of a cylinder, and you subtract the volume of the hollowed out part of the solid.
Volume of Washer = pi * (R^2 - r^2)h
(h is interchangeable with w whether you refer to it as width or height)
We know that f(x) = 4-x^2, 0.
We also know that 4-x^2 is the top most function while 0 is the bottom function, so we would take [4-x^2] - [0] and get 4-x^2 for the final f(x) function. Now, we plug and chug.
pi * PRETEND THERE IS AN INTEGRAL SIGN HERE [(4-x^2)^2] dx ---> 256pi/15
The washer method, like I said earlier, is a little more complex because you have to take into consideration the fact that part of the solid will be hollowed out. Essentially, you take the same equation for the disc method because it is still based off of the volume formula of a cylinder, and you subtract the volume of the hollowed out part of the solid.
Volume of Washer = pi * (R^2 - r^2)h
(h is interchangeable with w whether you refer to it as width or height)