Section 6.2 – Finding The Volume of Solids of Revolutions (Using Discs)

· In order to find the volume of the solid obtain by rotating the area between two curves, lines, etc, you must use integrals in order find the specified volume asked.

· When using it in math terms, you will take your equations and or lines and take whatever is on top squared (R²) minus whatever is on bottom squared (r²). For example, if you have the equationsand, is the top equation (R) and is the bottom equation (r) when graphed. This is used to determine the radius’ you will be using to calculate the area of the disk, using the area of a circle.

· Once you find this out, you set up your integral (top² – bottom²) or (R² - r²). If there is only one equation given in the problem, you just use R² as your integral.

· Since this time you are using discs to find the volume, rotating about a line or axis. Your limits depend on what you are rotating about. Your top limit being +r and bottom limit being –r.

· Remember since you are rotating about a line or axis, your integral will always be multiplied by Pi ().

For volume, the integral equation is:

à f(x)² is the top equation and g(x)² is the bottom equation. If there is only one equation in the problem then you do the integral only for f(x)².

After setting up the integral for the problem, you will do the anti-derivative of what’s inside. Once you have the anti-derivative, you will plug in your limits expressed (top limit first minus bottom limit second). After plugging in the limits, solving, and multiplying by Pi (), the volume between the curves, rotated about the axis or a line, is expressed.

Examples:

1) This problem has only one line and you are finding the volume of the solid formed by rotating the area bound by the x & y axis and the line.

-Here you are given your equation

Rotate about the x-axis to Generate the solid

-Solve for y to place as r for you interval

-integrate

2) With this problem, you have two different equations, therefore, you must figure out which equation represents R and which represents r by drawing a washer, not just a disc. R = line outside washer. r = line inside washer.

R

r

About the x-axis

This disc has a hole in the middle, making a washer

-Here you are given two equations. Using the washer method, determine which equation is r and which one is R.

-Set up your equation, integrate

3) When you are rotating your line to find the volume over a line other than the x or y-axis, you need to provide the missing section in your equation based on the graph.

About y=2

-Here are your two equations which you need to find R and r

-Set up your equation

-integrate

Here are some sites that may offer additional help:

http://tutorial.math.lamar.edu/AllBrowsers/2413/VolumeWithRings.asp

http://curvebank.calstatela.edu/volrev/volrev.htm