Volumes

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

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

Volumes

Prerequisite Knowledge

Geometry

Area

• Circle: $A=\pi r^2$$A=\pi r^2$, where $r$$r$ is the radius of the circle.

Volume

• Cylinder: $V=\pi r^{2}h$$V=\pi r^2h$, where $r$$r$ is the radius of the circular face, $h$$h$ is the height of the cylinder.
• Parallelepiped (3-dimensional box): $V=lwh$$V=lwh$, where $l$$l$, $w$$w$, and $h$$h$ are the lengths of the three sides.

Algebra

Fundamental Theorem of Algebra [FTA]

Every polynomial of degree $n$$n$ with complex coefficients has $n$$n$ roots in the complex numbers.

NOTE: We will only need to find the real roots of equations.

Definition: Discriminant

The discriminant to the quadratic equation $a{x}^2+bx+c=0$$ax^2+bx+c=0$ is ${\displaystyle b^2-4ac}$$\displaystyle b^2-4ac$.

Quadratic Formula

To find the solutions to $a{x}^2+bx+c=0$$ax^2+bx+c=0$, use ${\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}$$\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

If the discriminant is non-negative, i.e., $b^2-4ac\ge 0$$b^2-4ac\geq 0$ then the solutions, $x$$x$-values, are real numbers.

Solving Equations

1. Quadratic functions:

   • use Quadratic Formula or use method below for Polynomial functions

2. Polynomial functions:

   1. Move all terms to one side of the equation: $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_2{x}^2+a_{1}x+a_0=0$$a_nx^n+a_{n-1}x^{n-1}+\cdots + a_2x^2+a_1x+a_0=0$
   2. Factor into linear factors, if possible: $(x-b)(x-c)\cdots (x-p)=0$$(x-b)(x-c)\cdots(x-p)=0$
   3. Use the Zero Product Property of Real Numbers, i.e., set each factor to $0$$0$ and solve for $x$$x$.
   4. Use FTA to check that you found all real solutions.

3. Root functions:

   • $\sqrt{\text{arg}(x)}=0$$\sqrt{\text{arg}(x)}=0$, when $\text{arg}(x)=0$$\text{arg}(x)=0$, for any function, $\text{arg}(x)$$\text{arg}(x)$.

4. Exponential functions:

   • Exponential functions cannot be equal to $0$$0$ unless the function is shifted vertically.

5. Logarithmic functions:

   • ${\log }_b\left(\text{arg}(x)\right)=0$$\log_b\left({\text{arg}(x)}\right)=0$, when $\text{arg}(x)=1$$\text{arg}(x)=1$, for any function, $\text{arg}(x)$$\text{arg}(x)$.

6. Trigonometric functions:

   • Use the unit circle to determine when a trig function is zero.

Find Intersections of Two Curves

Suppose, $y_1=f(x)$$y_1=f(x)$ and $y_2=g(x)$$y_2=g(x)$, then the curves intersect at the $x$$x$-values of the solutions to $f(x)=g(x)$$f(x)=g(x)$.

Suppose, $x_1=f(y)$$x_1=f(y)$ and $x_2=g(y)$$x_2=g(y)$, then the curves intersect at the $y$$y$-values of the solutions to $f(y)=g(y)$$f(y)=g(y)$.

Find Intercepts of Curves

$x$$x$-intercepts

Replace all $y$$y$-variable(s) with $0$$0$, then solve for $x$$x$.

$y$$y$-intercepts

Replace all $x$$x$-variable(s) with $0$$0$, then solve for $y$$y$.

Sketching Curves

NOTE: Sketching is not the same as graphing. When graphing a curve, precision is expected, so 'T'-charts or sketching points on the curve are normally required; however, when sketching graphs, you need only get an approximate graph of the curve.

General Method

1. Find $x$$x$-intercept(s) and $y$$y$-intercept(s).

2. Use known properties of known functions:

   1. Linear functions, $y=mx+b$$y=mx+b$, $ax+by+c=0$$ax+by+c=0$
   2. Quadratic functions, $y=a{x}^2+bx+c$$y=ax^2+bx+c$
   3. Polynomial functions, $y=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_2{x}^2+a_{1}x+a_0$$y=a_nx^n+a_{n-1}x^{n-1}+\cdots + a_2x^2+a_1x+a_0$
   4. Root functions, $y=\sqrt{x}$$y=\sqrt{x}$, $y=\sqrt[n]{x}$$y=\sqrt[n]{x}$, for $n>1$$n>1$
   5. Exponential functions, $y=a^x$$y=a^x$, $x=a^y$$x=a^y$
   6. Logarithmic functions, $y={\log }_b(x)$$y=\log_b{(x)}$, $x={\log }_b(y)$$x=\log_b{(y)}$
   7. Trigonometric functions

3. If necessary, exchange $x$$x$ and $y$$y$ variables, sketch, then reflect graph over the $y=x$$y=x$ line.

4. Use your knowledge of translations of curves.

5. Plot point(s), only as a last resort.

Practice 01

Find the exact values of all real, if any solutions to the following equations:

1. ${\displaystyle x^2-5x=6}$$\displaystyle x^2-5x=6$
2. ${\displaystyle x^3=2}$$\displaystyle x^3=2$
3. ${\displaystyle \sqrt{x+3}=7}$$\displaystyle \sqrt{x+3}=7$
4. ${\displaystyle \ln (x+1)=2}$$\displaystyle \ln{(x+1)}=2$
5. ${\displaystyle \sin (x)=\frac{-\sqrt{3}}{2}}$$\displaystyle \sin{(x)}=\frac{-\sqrt{3}}{2}$
6. ${\displaystyle \sin (x)=2}$$\displaystyle \sin{(x)}=2$
7. ${\displaystyle {\tan }^{-1}(x)=2}$$\displaystyle \tan^{-1}{(x)}=2$
8. ${\displaystyle e^{x-7}=4}$$\displaystyle e^{x-7}=4$
9. ${\displaystyle 3x^3-x^2-3x+x=0}$$\displaystyle 3x^3-x^2-3x+x=0$
10. ${\displaystyle x^2=3}$$\displaystyle x^2=3$

Check Your Work

Use Technology to Solve Equations

Mathematica

​x
1
(* Example from Practice 01-2: Solve x^3=2 *)
2
(* NOTE: == is used for the equals sign in an equation *)
3
Solve[x^3 == 2, x, Reals]
4

5
(* Example from Practice 01-4: Solve ln(x+1)=2 *)
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(* NOTE: == is used for the equals sign in an equation *)
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Solve[Log[x+1] == 2, x, Reals]
8

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(* Example from Practice 01-5: Solve sin(x)=-sqrt{3}/2 *)
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(* NOTE: == is used for the equals sign in an equation *)
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Solve[{Sin[x] == -Sqrt[3]/2, 0 <= x < 2Pi}]
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(* or *)
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Reduce[{Sin[x] == -Sqrt[3]/2, 0 <= x < 2Pi}]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.
2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
3. When solve trigonometric equations, Mathematica output may include the symbol $\vee$$\or$ which is the mathematical abbreviation for “or.”

Practice 02

Find the exact values of all real intersection points:

1. ${\displaystyle y=x^2}$$\displaystyle y=x^2$; ${\displaystyle y=3+x}$$\displaystyle y=3+x$
2. ${\displaystyle y=2-x}$$\displaystyle y=2-x$; ${\displaystyle y=x^2+1}$$\displaystyle y=x^2+1$
3. ${\displaystyle y=\sin (x)}$$\displaystyle y=\sin{(x)}$; ${\displaystyle y=\frac{1}{2}}$$\displaystyle y=\frac{1}{2}$
4. ${\displaystyle x=y^2-2}$$\displaystyle x=y^2-2$; ${\displaystyle x=y}$$\displaystyle x=y$
5. ${\displaystyle x=\sqrt{y}}$$\displaystyle x=\sqrt{y}$; ${\displaystyle x=y}$$\displaystyle x=y$

Check Your Work

Use Technology to Find Intersections of Two Curves

Mathematica

xxxxxxxxxx
9
1
(* Example from Practice 02-4: Find intersection(s) of x=y^3-1 and x=y *)
2
(* NOTE: == is used for the equals sign in an equation *)
3
Solve[{x == y^2-2, x == y}, {x, y}, Reals]
4

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(* Example from Practice 02-3: Find intersections of y=sin(x) and y=1/2 *)
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(* NOTE: == is used for the equals sign in an equation *)
7
Solve[{Sin[x] == 1/2, 0 <= x < 2Pi}]
8
(* or *)
9
Reduce[{Sin[x] == 1/2, 0 <= x < 2Pi}]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.
2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
3. When solve for intersections of trigonometric equations, Mathematica output may include the symbol $\vee$$\or$ which is the mathematical abbreviation for “or.”

Practice 03

Sketch a graph of the following:

1. ${\displaystyle y=x^3+x^2-2x}$$\displaystyle y=x^3+x^2-2x$
2. ${\displaystyle y=\sqrt[3]{x}}$$\displaystyle y=\sqrt[3]{x}$
3. ${\displaystyle y=\sqrt{x+5}}$$\displaystyle y=\sqrt{x+5}$
4. ${\displaystyle y=\ln (x+2)}$$\displaystyle y=\ln{(x+2)}$
5. ${\displaystyle y=e^{-x}}$$\displaystyle y=e^{-x}$
6. ${\displaystyle y={\tan }^{-1}(x)}$$\displaystyle y=\tan^{-1}{(x)}$
7. ${\displaystyle x=y^3-2y^2}$$\displaystyle x=y^3-2y^2$

Check Your Work

Use Technology to Graph Curves

DESMOS

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. For built-in DESMOS functions, click on the keyboard icon on the bottom-left of the page.
2. DESMOS can graph more than one curve and can graph curves that are not functions.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Monday, 6 September 2020 13:33 EDT