vader1990

ok, its volume generated when the solid is revolved around stuff.

so, when its revolved about the x-axis or modification:

about y=a
its integral of a-f(x)
about y=-a
its integral of a+f(x)

ABOUT y-axis; using shells

about x=a
integral of (a-x)*f(x)
about x=-a
integral of (a+x)*f(x)

thanks a lot guys!!!

NYCDreamer89

Remember that the integrand of your solid of revolution formula (washer method) would be
f(x)^2 - g(x)^2

Where f(x) is your outside curve, and g(x) is your inside curve. In the case of revolving about a line parallel to the x axis, g(x)=constant= a. Also, f(x) > g(x) for all x between the bounds of your definite integral.

So I suppose it depends whether y=a is above or below f(x). If it is below f(x) then the integrand should be f(x)^2 - a^2. (Someone else check my logic :-p)

vader1990

nevermind I figured it out:

here's the whole shebang:

washers: about x-axis or horizontal lines:

about y=a (if a is ab above f and g)
v = pi * Integral ((a-BOTTOM graph)^2 - (a-TOP graph)^2)

about y=-a (if a is below f and g)
v = pi * Integral ((a + TOP graph)^2 - (a + BOTTOM graph)^2)

DISK METHOD: same as above BUT for y=a USE (a-TOP)^2 -(a-BOTTOM)^2 INSTEAD of bottom - top

SHELLS: about y-axis, and vertical lines:

x=a
v= 2pi * Integral ( (a-x)(top - bottom))

x=-a
v = 2pi * Integral( (a+x) * (top-bottom) )

THESE ARE ALL CORRECT; I HAVE TESTED THEM HEAVILY, BY USING THEM TO SOLVE QUESTION IN MY TEXT AND BARRONS AP GUIDE 2008