Horizontal Slicing

We can also compute areas by slicing horizontally instead of vertically.  In this case, we'll denote by $W(y)$ the width of a horizontal strip. The area of each slice is roughly $W(y)\cdot\Delta y$, and the total area becomes $\int_c^d W(y)\,dy$, where $[c,d]$ is the $y$-interval that includes the total area to be computed.  This means the horizontal strips start at the lowest $y$-value $c$, and end at the largest $y$-value $d$ -- these are not $x$-values here. 

Let's look at an example.  We have two curves below, $f$ and $g$, which are written in terms of $y$.  As in the case of vertical strips, we need to know which is the larger.  That translates to the curve further to the right (bigger $x$-values) being the larger.