Frozen Pages/Volume of Revolution

From Math Images

Solid of revolution

Field: Calculus
Image Created By: Lizah Masis
Website: [1]

Solid of revolution

This image shows the solid formed after revolving the region bounded by $y=x^2$ , $y=0$ , $x=0$ and $x=1$

Basic Description

When finding the volume of revolution of solids, in many cases the problem is not with the calculus, but with actually visualizing the solid. To find the volume of a solid like a cylinder, usually we use the formula ${\pi} {r^2} h$ . Alternatively we can imagine chopping up the cylinder into thin cylindrical plates, much like slicing up bread, computing the volume of each slice, then summing up the volumes of all the slices.

The disc method is much like slicing up bread and computing the volume of each slice http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html

This is the idea behind computing the volume of solids whose shapes are complicated. If you are given a function which decribes the shape of a solid, plot the function, then revolve the resultant plane area about a straight line to obtain the original solid, now called the the solid of revolution. The volume of the solid can then be computed using the disc method.
Note: There are other ways of computing the volumes of complicated solids other than the disc method.

http://mathdemos.gcsu.edu/mathdemos/diskmethod/diskmethod.html

In the disc method, we imagine chopping up the solid into thin cylindrical plates, each

${\Delta x }$ units thick, calculating the volume of each plate, then finding the Riemann sum of the volumes of all plates.

For example, let's consider a region bounded by $y=x^2$ , $y=0$ , $x=0$ and $x=1$

<-------Plotting this area,

If we revolve this area about the x axis ( $y=0$ ), then we get the solid on the right hand side of the page

A plane area being revolved http://curvebank.calstatela.edu/volrev/volrev.htm

To find the volume of the solid using the disc method:

Volume of one disc = ${\pi} y^2{\Delta x}$ where $y$ - which is the function- is the radius of the circular cross-section and $\Delta x$ is the thickness of each disc

To calculate the volume of all dics, we need to find the Riemann sum of all plates:

Volume of all discs = ${\sum}{\pi}y^2{\Delta x}$ , with $X$ ranging from 0 to 1

This is also the same as:

$\int_0^1 {\pi}y^2\,dx ={\pi}\int_0^1 (x^2)^2\, dx$

Evaluating this intergral,

[show][hide]

= ${\pi}\int_0^1 x^4 dx$

= $[{{x^5\over 5} + C|}_0^1] {\pi}$

= $[{1\over 5} + {0\over 5}] {\pi}$

= ${\pi}\over 5$

volume of solid= ${\pi\over 5} units^3$

If you are able, please consider adding to or editing this page!
Have questions about the image or the explanations on this page?
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.

Retrieved from "http://mathforum.org/mathimages/index.php/Frozen_Pages/Volume_of_Revolution"

Categories: High School | Math Images In Progress | Calculus | Lizah Masis