In these applications, all area is counted positively. If we slice horizontally and want the unsigned area between two functions from a to b we need to evaluate \displaystyle \int_a^b |f(y)-g(y)| \, dy . If you are using a numerical tool, you can just evaluate this integral using the absolute value of the integrand. In other cases if the curves for the two functions cross in the interval from a to b, it may be easier to divide the interval up into parts where the curves don't cross and evaluate each separately.
Think of each slice as being dy tall and the length is then the difference between f (y) and g(y).
The first example shown is \displaystyle \int_0^1 \sqrt{y}-0 \, dy . Move the y slider to move the sample rectangle.
The second example is \displaystyle \int_0^1 y^2- y \, dy .