AP Calculus AB and BC glossary

Region bounded by curves

Also called: Region between two curves

A region bounded by curves is the set of points enclosed between two graphs over an interval. Its area is the integral of upper minus lower in x, or right minus left in y. When the curves alone close the region, their intersection points give the limits; otherwise the named boundary lines do.

A = \int_{a}^{b} \left[f(x) - g(x)\right]\,dx \quad \text{when } f(x) \ge g(x)

The height of a vertical strip cut through the region is the top curve minus the bottom curve, $f(x) - g(x)$ . That difference is a length rather than a signed area, which is why the formula still works when the whole region sits below the $x$ -axis: the vertical distance between the curves is positive whenever $f$ lies above $g$ , whatever sign the two values carry. Where $f = -1$ and $g = -3$ the strip is still 2 units tall.

A = \int_{c}^{d} \left[p(y) - q(y)\right]\,dy \quad \text{when } p(y) \ge q(y)

That is the horizontal-strip version, with $p(y)$ the right boundary of the region, $q(y)$ the left, and $c$ and $d$ the $y$ -values where the region starts and stops. Switch to it when it makes the boundaries simpler. If the top of the region changes from one curve to another partway across, integrating in $x$ takes two integrals while integrating in $y$ may take one. Note also that the intersection points are the limits only when the curves alone close the region. If the problem also names vertical lines such as $x = 0$ , those lines set the limits instead.

The mistake

Integrating $f(x) - g(x)$ straight across an interval where the curves cross. Past the crossing point the difference turns negative and silently cancels part of the area already counted. Find every intersection first, split there, and write whichever curve is actually on top first in each piece.

Appears in: Unit 8: Applications of Integration