Volumes

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 6.2 – Volumes
• Section 6.3 – Volumes by Cylindrical Shells

Expected Educational Results

• Objective 11–1: I can set up the integral to compute the volume obtained by rotating a region about an axis.
• Objective 11–2: I can evaluate the integral to compute the volume obtained by rotating a region about an axis using FTC-II.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Volumes

Methods

Definition: Volume by Disk Method

The area $A$$A$ of the region bounded by the curves $y=f(x)$$y=f(x)$ and $y=g(x)$$y=g(x)$ where $f$$f$ and $g$$g$ are continuous and $f(x)\ge g(x)\ge d$$f(x)\geq g(x)\geq d$ for all $x$$x$ in $[a,b]$$[a,b]$ rotated about $y=d$$y=d$ is $V=\pi {\int }_a^b{R}^2(x)dx$$V=\pi \int_a^b {R^2(x)\,dx}$ where $R(x)=f(x)-d$$R(x)=f(x)-d$.

Definition: Volume by Washer Method

The area $A$$A$ of the region bounded by the curves $y=f(x)$$y=f(x)$ and $y=g(x)$$y=g(x)$ where $f$$f$ and $g$$g$ are continuous and $f(x)\ge g(x)\ge d$$f(x)\geq g(x)\geq d$ for all $x$$x$ in $[a,b]$$[a,b]$ rotated about $y=d$$y=d$ is $V=\pi {\int }_a^b[R^2(x)-r^2(x)]dx$$V=\pi \int_a^b{\Big[R^2(x)-r^2(x)\Big]\,dx}$ where $R(x)=f(x)-d$$R(x)=f(x)-d$ and $r(x)=g(x)-d$$r(x)=g(x)-d$.

Definition: Volume by Cylindrical Shells

The area $A$$A$ of the region bounded by the curves $y=f(x)$$y=f(x)$ and $y=g(x)$$y=g(x)$ where $f$$f$ and $g$$g$ are continuous and $f(x)\ge g(x)$$f(x)\geq g(x)$ for all $x$$x$ in $[a,b]$$[a,b]$ rotated about $x=c$$x=c$ is $V=2\pi {\int }_a^{b}p(x)h(x)dx$$V=2\pi \int_a^b{p(x)h(x)\,dx}$ where $p(x)=|x-c|$$p(x)=|x-c|$ and $h(x)=f(x)-g(x)$$h(x)=f(x)-g(x)$.

Procedure

1. Sketch both curves.

2. Identify the region enclosed by both curves.

3. Sketch a typical rectangle within the region where every potential rectangle starts and ends at different curves:

   

4. Identify the axis of rotation in the graph (use dotted line to quickly identify this axis in your drawing).

5. Find intersection(s) of curves, by

   1. Setting $f(x)=g(x)$$f(x)=g(x)$ and solve for all $x$$x$-values [when using vertical rectangles], OR
   2. Setting $f(y)=g(y)$$f(y)=g(y)$ and solve for all $y$$y$-values [when using horizontal rectangles]

6. Identify the method (dashed lines in the following pics are axes of rotation). Explain.

   

7. Set up the integral.

8. Evaluate the integral.

Investigation 08

1. Use any software program to estimate the volume of the solid obtained by rotating about the $x$$x$-axis the region bounded by $y=e^{-x^2}$$y=e^{-x^2}$ and the $x$$x$-axis on $[0,2]$$[0,2]$.
2. Use any software program to estimate the volume of the solid obtained by rotating about $y=4$$y=4$ the region bounded by $y=\ln (x)$$y=\ln{(x)}$ and $y=x-6$$y=x-6$.

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Last Modified: Tuesday, 29 September 2020 11:49 EDT