Volume with Cross Sections
Part 2

This lesson is part 2 of a 3 part series for how to find the volume of a solid with known cross‑sections using definite integrals. The key focus is to express the area of each slice as a function of x or y and integrate it to obtain the total volume.

Summary

In this video, the base region R is bounded by the graphs of f(x) and g(x). First we find the intersection points of f and g, either algebraically or using Desmos, and label them a and b. Then, for several different types of cross‑sections—rectangles with height twice the width, squares with base in R, squares with diagonal in R, equilateral triangles, and semicircles—we walk through how to build an area formula and integrate from a to b with respect to x. Desmos is used to check or approximate the numerical values of the resulting volumes.

Key Takeaways

• The base region R is bounded by the curves y = f(x) and y = g(x), and the vertical boundaries are x = a and x = b, found by solving f(x) = g(x).

• For any vertical slice, the base length in R is top minus bottom: g(x) − f(x).

• Rectangular cross‑sections with height twice the width have area A(x) = 2[g(x) − f(x)]², because height = 2·(g(x) − f(x)).

• Square cross‑sections with base in R have side length s = g(x) − f(x), so A(x) = [g(x) − f(x)]².

• Square cross‑sections with diagonal in R use diagonal d = g(x) − f(x), side s = d/√2, and area A(x) = d²/2 = (g(x) − f(x))² / 2.

• Equilateral triangle cross‑sections with side in R have side s = g(x) − f(x) and area A(x) = (√3/4)[g(x) − f(x)]².

• Semicircular cross‑sections with diameter in R have diameter d = g(x) − f(x), radius r = d/2, and area A(x) = (1/2)πr² = (π/8)[g(x) − f(x)]².

• In every case, the volume is found by integrating the area function: V = ∫A(x) dx, and the numerical value can be approximated with a calculator or Desmos.

Transcript

Introduction

All right guys, Mr. Antonucci here, and we are back to looking at how to find the volume of a solid with known cross-sections.

In this example, we have a region R bounded by the graphs of f and g, and I am going to show you how to do a mix of this in Desmos as well as on paper.