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### Course: AP®︎/College Calculus AB>Unit 8

Lesson 9: Volume with disc method: revolving around x- or y-axis

# Disc method around y-axis

Finding the volume of a figure that is rotated around the y-axis using the disc method. Created by Sal Khan.

## Want to join the conversation?

• How will I know if I'll use the disk, ring or shell method? And when do I use vertical or horizontal?
• It really depends on the situation you have. If you have a function y=f(x) and you rotate it about the x axis, you should use disk (or ring, same thing in my mind). If you rotate y=f(x) about the y axis, you should use shell.
Of course, you can always use both methods if you can find the inverse of the function. If I wanted to rotate y=x^2 about the y axis, that would be equivalent to rotating x=√(y) about the x axis. I prefer to not bother with finding the inverse of the function. I would just identify the situation and use the appropriate method.
By the way, if you have a function that is reverse, that is to say x=f(y), everything I said would be reversed. You would use disk about the y axis and shell about the x axis.
• Anyone that might explain why Sal takes the principle root of y at ? For some reason it is completely lost on me..
• He did that because he is rotating around the y-axis and he is trying to find the volume. As Sal showed, you need to find the radius of each disk so as to apply it into A = (pi)r^2 and then V = A(dy). Notice that it is in terms of dy, not dx. Therefore, the equation y=x^2 needed to be changed into terms of x, otherwise you would be finding a radius and thus an area and thus a volume of a solid that is irrelevant to this problem.
• At , why did we use the interval from 1 to 4 instead of from 0 to 4?
• See - it is 1 to 4 because that is how he decided to define the shape of the object whose volume is being measured.
• What are the practical uses of the disk method (or finding the volume of solids of revolution in general)? What are some physical applications of these problems?
• Edison's volume or displacement techniques won't work to estimate the volume of material in coronal loops (http://news.discovery.com/space/sun-throws-huge-and-beatiful-plasma-arc-into-space-140925.htm), the volume of brick in underwater tels showing a massive ancient flood in the Black Sea, or the volume of seamounts (http://en.wikipedia.org/wiki/Hawaiian%E2%80%93Emperor_seamount_chain).
Newton wrote calculus and physics in the same book as one idea. The book's title "Il Principia" means "The Idea." Whenever you wonder an application, for calculus, think "physics." Whenever the math in a physics question seems inefficient, try attempting the same question, after completing first-year calculus. Sometimes second-year calculus is required. When you have the math, physics is pretty simple.
• The integral used in the problem is in terms of y. However, since y=x^2 and thus dy=2x dx, can you redefine the integral so that it is from 1 to 2 and its expression is 2pix^3 dx?
• That's a very creative approach. It seems to work, but involves substituting things around and updating the bounds of integration. You still have to solve for x in terms of y (the square root bit) to transform the bounds from y to x and there is a little more simplification required at the end. This adds points at which stupid algebra errors can creep in.
If it helps you get the right answer and it does give you the right answer every time, go for it. Though you may have to argue with your teacher for a couple of "show your work" points because you do it differently. I leave it to someone more experienced to show proof.
• is the volume always positive?
• Volume is defined as the amount of space inside a solid. This is always a positive number because an object cannot take up a negative amount of space. Hope this helps!
• What if we rotate it around any other line, say, y = x? How would we find out the volume in that case?

And if we can rotate around any line, would it not be so bizarre as to think of rotating it just about any function, be it quadratic or anything else? What would we do in such a scenario to find the volume?
• how do you decide the limits of the integral?
• It will be stated if there is one curve and/or they will be the intercepts of two curves