Chapter 6: Applications of Definite Integrals

Section 6.1: Volumes Using Cross-Sections

This section explores how definite integrals can be used to find the volumes of solids with known cross-sectional areas. The method involves integrating the area of a cross-section, taken perpendicular to a given axis, across the bounds of the solid.

Finding Volume by Integrating Cross-Sectional Areas

• Key Concept: The volume of a solid can be found by integrating the area of its cross-sections perpendicular to an axis.

• Formula: If A(x) is the area of the cross-section at position x, then the volume V is given by:

• Interpretation: Integrating a length gives an area; integrating an area gives a volume.

Example 1: Volume of a Pyramid

• Problem: Find the volume of a pyramid 3 m high with a square base 3 m on each side.

• Solution Outline:

  • Let x be the distance from the vertex along the height.

  • The cross-section at x is a square. Its side length is proportional to x.

  • Area of cross-section:

  • Set up the integral for volume using the limits from 0 to 3.

  • Evaluate the integral to find the volume.

• General Formula for a Pyramid:

Example 2: Solid with Equilateral Triangle Cross-Sections

• Problem: The base is bounded by the x-axis and the curve . Cross-sections perpendicular to the x-axis are equilateral triangles.

• Solution Outline:

  • Base of triangle at x:

  • Area of equilateral triangle:

  • Volume: over the interval where

Justification of the Method

• Partition the interval [a, b] into n subintervals of width .

• Approximate the volume by summing the volumes of thin slabs:

• As , this sum approaches the exact volume:

Solids of Revolution: The Disk Method

When a region is revolved around an axis, the resulting solid's volume can be found using the disk method if the cross-sections perpendicular to the axis are disks.

• Area of a Disk:

• Volume:

Example: Volume of a Sphere

• Region: Above the x-axis, below , revolved about the x-axis.

• Volume:

Solids of Revolution: The Washer Method

If the region being revolved is not bounded by the axis, the cross-sections are washers (annuli). The washer method subtracts the inner radius from the outer radius.

• Area of a Washer:

• Volume:

Example: Washer Method with y-Axis Revolution

• Find the volume of the solid created by revolving a region about the y-axis.

• Outer radius: , Inner radius:

• Area:

• Volume:

Cavalieri’s Principle

• Statement: Solids with equal altitudes and identical cross-sectional areas at each height have the same volume.

• This principle justifies the use of cross-sectional area integration for finding volumes.

Practice Problems

1. Find the volume of the solid whose base is the region bounded by the x-axis, the y-axis, and . The cross-sections perpendicular to the x-axis are semicircles with diameters running across the base of the solid.

2. Find the volume of the solid generated by revolving the region bounded by , the y-axis, , and about the y-axis.

3. Find the volume of the solid generated by revolving the region bounded by and about:

   • (a) the x-axis

   • (b) the line

Additional info: The images included above are directly relevant to the explanation of the washer method and its application to solids of revolution, as they visually clarify the geometric setup and the meaning of outer and inner radii in the context of the definite integral for volume.