Determine the volume of the solid generated by revolving the region enclosed by the circle x^2 + y^2 = 1 about the line x = -2.The solid resembles a doughnut.
Post solutions, please. :)
2007-09-17 03:03:34 · 7 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
You can use Pappus centroid theorem: the volume of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area of the lamina and the distance traveled by the lamina's geometric centroid.
Here, lamina means any closed region and centroid is the the same as center of gravity (or center of mass).
Suppose the circle has radius r and revolves about a line whose distance to the the center of the circle (it's center of gravity) is d >=r. The circle has area A = pi r^2 and it's center travels the distance 2pi d (it describes a circle of radius d). So, the volume of the solid generated by such revolution is
V = pir^2 * 2pi d = 2 (pi r)^2 d. (cubic units)
In your case, the circle has radius 1 and the line x = -2 is at distance 2 from the center. So, the volume is
V = 2 (pi)^2 2 = 4 pi^2 cubic units
2007-09-17 03:27:04 · answer #1 · answered by Steiner 7 · 0⤊ 1⤋
That is a standard geometrical problem.
1. Draw a graph to make yourself a better picture of the situation. Use: y=squrt(1-x^2) and draw the line x=-2 too.
2. The simple form you call a doughnut has a mathematical term too. A Torus.
3. The equation you need is:
V=2*pi^2*R*r^2=2*3,41^2*2=39,47
Your Torus has 39,5 Volume units!
Was a pleasure
2007-09-17 03:29:00 · answer #2 · answered by Marcus Paul 3 · 0⤊ 0⤋
x^2 + y^2 = 1 is a unit circle about the origin.
Calculus is not strictly necessary in this case; you can use "Pappus' Centroid Theorem."
"The volume of any solid of revolution, is equal to the area of the lamina which generates the solid, multiplied by the distance traveled by the lamina's centroid."
V = 2π dc A,
(where dc is the distance of the centroid from the axis, and A is the area of the lamina.)
A unit circle has an area of π square units and it's centroid is at the origin, so the distance from the axis is
dc = 2
that means the solid has a total volume of 4π² cubic units...
Hope that makes sense,
~W.O.M.B.A.T.
2007-09-17 03:25:59 · answer #3 · answered by WOMBAT, Manliness Expert 7 · 0⤊ 0⤋
side S = 10 x = square cut on each corner S - 2x . . . is the remaining side . . . folding the board V = ( S - 2x )² x = ( 10 - 2x )² x. . . . volume of the box V = ( 100 - 40x + 4 x² ) x = ( 100 x - 40x² + 4 x^3 ) . . . differentiating dV/dx = 100 - 80x + 12 x² . . . equating to zero 100 - 80x + 12 x² = 0 x = 5 / 3 volume = ( 10 - 2x )² x = ( 10 - 10 / 3 )² 5 / 3 = (20 / 3)² ( 5 / 3 ) Volume = 74.074
2016-05-17 04:44:51 · answer #4 · answered by Anonymous · 0⤊ 0⤋
there is a formula to calculate the volume of a rotating object .
but i cannot put the term back into English.
volume= (area of the closed curve) * (the length of the orbit passed by the centroid of the closed curve)
that is
volume=pi*1^2 *(2*pi*2) = 4pi^2
2007-09-17 03:24:42 · answer #5 · answered by Anonymous · 0⤊ 0⤋
This one is --such-- a classic that I'm not gonna spoil it for you ☺
I'm sure that they talk about 'volumes of revolution' in your textbook. Just read up on the method and then do it.
Doug
2007-09-17 03:16:51 · answer #6 · answered by doug_donaghue 7 · 2⤊ 0⤋
http://whistleralley.com/torus/torus.htm
2007-09-17 03:07:10 · answer #7 · answered by gjmb1960 7 · 0⤊ 0⤋