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.

In this example, $W(y)=f(y)-g(y)$. We compute our
area:$$\int_c^d
W(y)\,dy=\int_{-1}^1\left((f(y)-g(y)\right)\,dy=\int_{-1}^1(1-y^2-(y^3-y))\,dy=\int_{-1}^1(1-y^2-y^3+y)\,dy.$$
DO: Before looking ahead,
evaluate this integral. Check your antiderivative!

In the example above, the two curves were naturally written as
functions of $y$, since they are not functions of $x$ (why not?). However, we often are
given functions of $x$, and we have to find the inverse function to
get a function of $y$. You will see this in the video.
What we do when we are given $y=f(x)$ is we find $f^{-1}$, since
$x=f^{-1}(y)$.