# Volume of revolution on an area crossing the axis

Suppose a question asks for the volume of revolution about the x axis to be found on a piece of area enclosed between 2 graphs, where the area crosses the x-axis. In this case, the method involving the subtraction of two volumes does not seem to work as there is overlap due to the area crossing the x axis. How would this volume be calculated?

• Use the revolution of max(f(x),-f(x)). – coffeemath Sep 2 '15 at 19:29
• @coffeemath What about for surface area please? – user636310 Mar 1 '19 at 12:37

Suppose you have two functions $y_1$ and $y_2$ suitably well-behaved over an interval $[a,b]$ and you want the total volume created by rotating the area between the graphs about the $x$ axis. Suppose also that either graph could cross the $x$ or intersect with the other.

You would need to split the domain of integration into contiguous sub-intervals determined by the solutions of the equations $y_1=0$, $y_2=0$ and $|y_1|=|y_2|$.

If for any such interval $[p,q]$ we have $y_i\geq y_j\geq 0$ or $y_i\leq y_j\leq 0$, you need to calculate $$\pi\int_p^q(y_i^2-y_j^2)dx$$

And if for any such interval $[r,s]$ we have $y_i$ and $y_j$ have opposite parity, you need to determine which has larger magnitude. If this is $y_i$ for this interval, you only need $$\pi\int_r^s y_i^2 dx$$