Concept preview

What you will be able to do

1. 01Formulate the area between two curves as $A=\int_a^b(\text{top}-\text{bottom})\,dx$, identifying which function is on top over the interval.
2. 02Find the bounds of integration by solving for the intersection points of the two curves, and verify which curve is greater on the enclosed region.
3. 03Handle regions where the curves cross by splitting the interval or using $\int|\text{top}-\text{bottom}|\,dx$, so the computed area is always positive.

Picture two curves, $y=f(x)$ on top and $y=g(x)$ on the bottom, enclosing a region between $x=a$ and $x=b$. Slice the region into thin vertical strips of width $dx$. Each strip is a tiny rectangle whose height is the vertical distance between the curves at that $x$, namely $f(x)-g(x)$ (top minus bottom), and whose width is $dx$. Its area is $\big(f(x)-g(x)\big)\,dx$. Adding up all the strips and taking the limit is exactly a definite integral: $$A=\int_a^b\big(f(x)-g(x)\big)\,dx = \int_a^b(\text{top}-\text{bottom})\,dx.$$