Volumes Using Cross-Sections

Definition

The volume $V$ of a solid of integrable cross-sectional area $A(x)$ from $x = a$ to $x = b$ is the integral of $A$ over that interval: $V = \int_{a}^{b} A(x) \, dx$

• How to read: “V equals integral from a to b of A of x dx.”
• Meaning: Slice the solid perpendicular to the $x$-axis, sum areas $A(x)$ times thickness $dx$.

Why It Matters

Integration by cross-sections is the generalized foundation of all volume measurement. It allows us to calculate the capacity of anything from a heart chamber to a cooling tower, proving that any 3D object can be understood as an accumulation of 2D slices.

Core Concepts

• Cross-Sectional Area Function: $A(x)$ represents the area of a slice of the solid perpendicular to the axis of integration.
  • How to read: “A of x.”
  • Meaning / when to use: Cross-sectional area at position $x$; derive $A(x)$ from slice geometry (circle, rectangle, triangle) before integrating.
• Integrability: The method assumes the cross-sectional area function is integrable over the given interval $[a, b]$.
• Summation of Slabs: The volume is conceived as the limit of Riemann sums of thin slabs with volume $A(x) \Delta x$.
  • How to read: “A of x times delta x.”
  • Meaning: Each slab is a thin slice of area $A(x)$ and thickness $\Delta x$; the integral is the limit as $\Delta x \to 0$.

Connected Concepts

• Conic Sections in Polar Coordinates
• Conic Sections
• Cross Product