Volume with Cross Sections
Part 1

This lesson is part 1 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

The video walks through two examples: one using square cross‑sections and another using equilateral triangles over a circular base. These examples show how to set up area formulas, determine limits of integration, and apply the general volume formula. Students use this method whenever a solid has a base in the plane and a specified shape for each perpendicular cross‑section.

Key Takeaways

• Volume of a solid with known cross‑sections is found by integrating the cross‑sectional area.

• The general formula is V = ∫ A(x) dx or V = ∫A(y) dy.

• Cross‑sections must have a known area formula such as square, rectangle, triangle, or semicircle.

• You choose the variable of integration based on the direction of the slicing.

Transcript

Introduction

All right guys, Mr. Antonucci here. In this section we are talking about the volume of a solid with known cross‑sections and how to find that volume. Previously we looked at the disc method and the washer method, which use circular cross‑sections, but we are not limited to circles. You can use this idea for any shape as long as you know the area formula for the cross‑section. Common shapes include squares, rectangles, triangles, and semicircles. The idea is to cut the solid into thin slices perpendicular to an axis and add their volumes. This works because each cross-section’s area can be written as a function of position.