AP Calculus AB and BC glossary

Cross-section

Also called: Known cross-sections

A cross-section is the two-dimensional shape you get by slicing a solid perpendicular to an axis. If you know the area of each cross-section as a function of position, integrating that area function along the axis gives the volume.

V = \int_a^b A(x)\,dx

The whole problem is writing $A(x)$ . For squares built on a base of width $w$ the area is $w^2$ ; for semicircles of diameter $w$ it is $\frac{\pi w^2}{8}$ ; for equilateral triangles of side $w$ it is $\frac{\sqrt{3}}{4}w^2$ .

Diameter or radius

When cross-sections are semicircles the base segment is the diameter, so the radius is half of it. Substituting the full width as the radius is the classic error and quadruples the answer.

Appears in: Unit 8: Applications of Integration