Calculus Integration Test: Volumes of Solids

Steve Phelps, Mar 1, 2014

This is a GeoGebraBook of various solids used on a Calculus test. Students were to use integration to find the volumes of the various solids.

Table of Contents

  1. Introduction
    1. Building Solids Introduction
  2. Cross Sections
    1. Cross-Sections Perpendicular to x-Axis: Squares
    2. Cross-Sections Perpendicular to x-Axis: Equilateral Triangle
    3. Cross-Sections Perpendicular to x-Axis: Right Triangles
    4. Cross-Sections Perpendicular to x-Axis: Rectangles
  3. Solids of Revolution
    1. Solids of Revolution: Around the x-Axis
    2. Solids of Revolution: Around the y-Axis
    3. Solids of Revolution: Around y=6

1.

Introduction

  • 1. Building Solids Introduction

1.1.

Building Solids Introduction

In the applet below you will find the functions [math]f(x)=2cos(\frac{\pi}{4}(x-1))+3[/math] and [math]g(x)=2cos(\frac{\pi}{12}(x-1))+3[/math] with the region of interest between the two curves shaded. We will use this region to construct a variety of different solids.
Building Solids Introduction
– Steve Phelps

2.

Cross Sections

  • 1. Cross-Sections Perpendicular to x-Axis: Squares
  • 2. Cross-Sections Perpendicular to x-Axis: Equilateral Triangle
  • 3. Cross-Sections Perpendicular to x-Axis: Right Triangles
  • 4. Cross-Sections Perpendicular to x-Axis: Rectangles

2.1.

Cross-Sections Perpendicular to x-Axis: Squares

The applet below shows the solid formed by erecting squares on cross-sections perpendicular to the x-axis.
Cross-Sections Perpendicular to x-Axis: Squares
– Steve Phelps

2.2.

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2.3.

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2.4.

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3.

Solids of Revolution

  • 1. Solids of Revolution: Around the x-Axis
  • 2. Solids of Revolution: Around the y-Axis
  • 3. Solids of Revolution: Around y=6

3.1.

Solids of Revolution: Around the x-Axis

The applet below shows the solid formed by rotating the region around the x-axis.
– Steve Phelps

3.2.

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3.3.

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Information