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6.2: Volume by Cross Sections

Difficulty Level: At Grade Created by: CK-12
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This activity is intended to supplement Calculus, Chapter 5, Lesson 2.

ID: 12280

Time Required: 15 minutes

Activity Overview

In this activity, students will be introduced to the concept of finding the volume of a solid formed by cross sections of a function that form certain shapes. Since volume is the area of the base times the height and dV=Areadx, students review areas of various shapes like squares, semicircles and equilateral triangles. Calculator screenshots are used to help students get a visual of the volume under consideration. Students will practice what they learn with exam-like questions.

Topic: Volume by Cross Sections

  • Applications of integration
  • Volume by cross sections

Teacher Preparation and Notes

  • Part 1 of this activity takes less than 15 minutes. Part 2 contains three exam-like questions that have accompanying visuals that can be used as an extension or homework.
  • Students will write their responses on the accompanying handout where space is provided for students to show work when applicable.

Associated Materials

Part 1 – Setting Up The Problem And Understanding The Concept

In this section students are introduced to the concept of finding the volume of a solid formed by cross sections of a function that form certain shapes. Since volume is the area of the base times the height and dV=Area dx, students review areas of various shapes like squares, semicircles and equilateral triangles.

Part 1 ends with students finding the volume with equilateral triangle cross sections.

Student Solutions

  1. dx
    1. base times height. The area of a square with side x is x2.
    2. 12πr2
  2. 12y32y
  3. 0.433013 cm2
  4. 0212y32 y dx=0212(xex2)32(xex2)dx=0234xe2x2dx

If students use usubstitution, u=2x2,du=4x dx and the limits of integration are from 0 to -8.

31608eudu=316(e81)=316(11e8)

Part 2 – Homework

This section enables students to get a visual of challenging exam-like questions. Students should show their work on the first two questions and show their set up on the third question.

Student Solutions

  1. 3π32 units3
  2. 2 units3
  3. 1.57 units3

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