6.1 Areas between Curves

In Calculus I we used the definite integral to find the area of a region lying between a curve and the $x$-axis. Very often, though, the region we care about is squeezed between two curves: the lens of glass between two arcs of a window, the gap between a demand curve and a market price, the difference between the water flowing into a reservoir and the water flowing out. None of these regions touches the $x$-axis in a helpful way.

The good news is that no new theory is needed — only the idea that made the definite integral work in the first place. We slice the region into thin strips, approximate each strip by a rectangle, add up the rectangles and take the limit. The only change is the height of each rectangle: instead of "curve minus axis" it becomes "top curve minus bottom curve". That single idea, slice–approximate–add–limit, will carry us through the whole of this chapter.

Area between Two Curves

Let $f$ and $g$ be continuous on $[a,b]$ with $f(x)\ge g(x)$ for every $x$ in $[a,b]$. Let $R$ be the region bounded above by $y=f(x)$, below by $y=g(x)$, and on the left and right by the vertical lines $x=a$ and $x=b$.

Divide $[a,b]$ into $n$ subintervals of equal width $\Delta x=\dfrac{b-a}{n}$ with end points $a=x_0\lt x_1\lt\cdots\lt x_n=b$, and choose a sample point $x_i^*$ in the $i$-th subinterval. Over that subinterval the thin strip of $R$ looks almost like a rectangle of width $\Delta x$ and height $f(x_i^*)-g(x_i^*)$ — the distance from the bottom curve up to the top curve. Adding the rectangles, $$A\approx\sum_{i=1}^{n}\left[f(x_i^*)-g(x_i^*)\right]\Delta x .$$ This is a Riemann sum for the continuous function $f-g$, so as $n\to\infty$ it converges to a definite integral.

Curves That Cross

Theorem 6.1 assumed that one curve stays on top. When the graphs cross, the top and bottom curves swap roles, and $\int_a^b[f(x)-g(x)]\,dx$ no longer measures area: on the pieces where $g$ is on top the integrand is negative and those pieces are subtracted. Area is never negative, so we use the distance between the curves, $|f(x)-g(x)|$: $$A=\int_a^b\left|f(x)-g(x)\right|dx .$$ In practice we find every crossing point $c$ in $(a,b)$, split the interval there, and on each piece integrate top minus bottom.

Integrating with Respect to $y$

Some regions are awkward to cut vertically. If the region is bounded on the right by $x=u(y)$ and on the left by $x=v(y)$, cut it into thin horizontal strips instead. A strip at height $y$ has length $u(y)-v(y)$ (right minus left) and thickness $\Delta y$, and the same slice–add–limit argument gives the following result.

Horizontal strips are worth trying whenever the vertical strips would change their top or bottom curve part-way across — each change costs a separate integral — or when the boundary curves are naturally written as $x$ in terms of $y$.

An Applied Problem: Consumer and Producer Surplus

Areas between curves appear whenever a quantity is the accumulated difference of two rates or two prices. A classic example comes from economics. A demand curve $p=D(q)$ gives the price buyers are willing to pay for the $q$-th unit; a supply curve $p=S(q)$ gives the price at which producers are willing to sell it. The market settles at the equilibrium $(q_e,p_e)$ where $D(q_e)=S(q_e)=p_e$.

The same picture describes many "gap between two rates" problems: if water flows into a tank at rate $r_{\text{in}}(t)$ and out at rate $r_{\text{out}}(t)$, the area between the two rate curves over $[t_1,t_2]$ is the net amount gained (where $r_{\text{in}}\ge r_{\text{out}}$); if two cyclists ride with velocities $v_A(t)\ge v_B(t)$, the area between the velocity curves is how far $A$ pulls ahead (Exercise III.3).

Exercises