Concept preview

What you will be able to do

1. 01Set up the area between two curves as a horizontal-strip integral with respect to $y$, $A=\displaystyle\int_c^d\left(\text{right}-\text{left}\right)\,dy$, when the curves are more naturally written as $x=g(y)$.
2. 02Choose deliberately between $dx$ and $dy$ by picking the variable whose strips avoid splitting the region, and convert an equation from $y=f(x)$ to $x=g(y)$ when that helps.
3. 03Find the $y$-bounds from the intersection points, decide which curve is the right boundary and which is the left, and evaluate the integral to a numerical area without reversing right and left.

Start from what you already trust. When you integrate a region with vertical strips, you write $A=\int_a^b\left(y_{\text{top}}-y_{\text{bottom}}\right)\,dx$: the strip is vertical, its thickness is $dx$, and its height is the upper curve minus the lower curve. The whole method rests on two facts — the strip has a definite top and a definite bottom, and both are single functions of $x$ across the whole interval $[a,\,b]$. When those facts hold, vertical strips win.