AP Calculus BC Practice Test 2026 - Free Calculus BC Practice Questions and Study Guide

Think of building the solid by stacking thin slices perpendicular to the x-axis. Each slice has cross-sectional area A(x) and thickness dx, so its volume is A(x) dx. Adding up all these slices over the interval from a to b gives the total volume, which is V = ∫_a^b A(x) dx. This uses the fundamental idea of slicing: area times thickness summed continuously along x yields volume.

The other expressions don’t match this idea. Multiplying by x would weight each slice by its position, giving a different quantity. Using dx^2 isn’t a standard infinitesimal for volume in this context. Using the derivative dA(x)/dx times dx would collapse to dA, which represents a change in area, not the actual volume formed by the slices.