Integration to find volume generated

[shyta]

To find the volume generated by rotating an area bounded by 2 curves f(x) and g(x) around the x-axis
we use the formula

Int of lower and upper intersections pi( f(x)^2-g(x)^2 ) dx

or

Int of lower and upper intersections pi (f(x)-g(x))^2 dx


I understand that they are different but i am confused when to use the correct formula.

 

[LCKurtz]

To find the volume generated by rotating an area bounded by 2 curves f(x) and g(x) around the x-axis
we use the formula

Int of lower and upper intersections pi( f(x)^2-g(x)^2 ) dx

or

Int of lower and upper intersections pi (f(x)-g(x))^2 dx


I understand that they are different but i am confused when to use the correct formula.

The first formula would be used if 0 ≤ g(x) ≤ f(x) on the interval [a,b] and the region is being revolved about the x axis. The second one doesn't look like it would ever be correct. If you were revolving the same type area around the y-axis with a >0 you would use the "shell" formula

$$\int_a^b 2\pi x(f(x) - g(x)) \ dx$$

and there are similar formulas for rotation about the y axis.