Volume with Cross Sections
Part 3

This lesson is part 3 of a 3 part series for how to find the volume of a solid with known cross‑sections using definite integrals. The key focus ishow to set up (and evaluate) volumes of solids with known cross-sections when the cross-sections are perpendicular to the y-axis. The key move is rewriting your boundary curves so x is written as a function of y, then building the cross-sectional area using “right minus left.”

Summary

In this video, you'll learn how to determine if the slicing direction is vertical or horizontal in a way that makes one variable more convenient. You’ll see how the same region can produce different volumes depending on whether the cross-sections are squares, semicircles, or rectangles.

Key Takeaways

• If cross-sections are perpendicular to the y-axis, you must integrate with respect to y (use dy).

• Rewrite each boundary curve into the form x = f(y) so you can measure widths horizontally.

• The cross-section “width” (or diameter/side) is usually right boundary − left boundary: g(y) − f(y).

• Find the y-limits of integration by solving for the intersection points of the two curves, giving y = c (lower limit) and y = d (upper limit).

Transcript

Introduction

 All right guys, Mr. Antonucci here, and we’re doing volumes of solids with known cross-sections. What we want to do is sketch the graph setup and evaluate the integral expression that represents the volume of the solid formed when the graphs form the region S