[size=100]Bases are equilateral triangles[br]Triangle Side = 3.5 cm[br]Prism height = 21 cm[/size]
Cross Sectional Specials (Calculus)
Here you will find several resources to help visualize solids formed by PRISM STACKING. For each resource here, you can modify the upper function, lower function, and lower & upper limits of integration (a and b, respectively). CROSS SECTIONS of any surface with any plane parallel to the yAxis are given in the individual resource title. Have fun creating and exploring!
Table of Contents
- Cross SEC = REG Polygon
- Modifiable Solid: Equilateral Triangle Cross Sections Parallel to yAxis
- Modifiable Solid: Square Cross Sections Parallel to yAxis
- Modifiable Solid: Regular Pentagon Cross Sections Parallel to yAxis
- Modifiable Solid: Regular Hexagon Cross Sections Parallel to yAxis
- Modifiable Solid: Regular Heptagon Cross Sections Parallel to yAxis
- Modifiable Solid: Regular Octagonal Cross Sections Parallel to yAxis
- Modifiable Solid: Regular Nonagon Cross Sections Parallel to yAxis
- Modifiable Solid: Regular Decagon Cross Sections Parallel to yAxis
- Other Types of Cross Sections
- Modifiable Solid: Rectangular Cross Sections Parallel to yAxis
- Modifiable Solid: Isosceles Right Triangle Cross Sections Parallel to yAxis (VA)
- Modifiable Solid: Isosceles Right Triangle Cross Sections Parallel to yAxis (VB)
- Modifiable Solid: Isosceles Triangle (Modifiable) Cross Sections Parallel to yAxis
- Modifiable Solid: Isosceles Trapezoid Cross Sections Parallel to yAxis
- Custom Solid: Changeable Rhombus Cross Section Parallel to yAxis
- Modifiable Solid: Semicircular Cross Sections Parallel to yAxis
- Modifiable Solid: Parabolic Cross Sections Parallel to yAxis
- Custom Solid: Elliptical Cross Sections Parallel to yAxis
- Custom Solid: Cross Sections Parallel to yAxis are Quadrilaterals Always Similar to the Base
- Prism Stacking Resources
- Building Surfaces with Cross Sections and Function Modeling
- Solids Formed From Certain Cross Sections
- Solids with Various Cross Sections: AR Templates
Modifiable Solid: Equilateral Triangle Cross Sections Parallel to yAxis
HOW TO USE:
Enter any [b]UPPER FUNCTION[/b] (labeled upper(x)).[br]Enter any [b]LOWER FUNCTION[/b] (labeled lower(x)).[br]Enter your [b]lower and upper limits of integration[/b] ("a" and "b", respectively).[br][br]Slide the [b]FILLING[/b] slider all the way to the left. [br]For this solid, note how cross sections parallel to the yAxis are[b] [color=#ff7700]equilateral triangles [/color][/b]with one of its sides lying in the base. This side has length = upper(x) - lower(x). [br][br]You can move this [color=#ff7700][b]equilateral triangle[/b][/color] by sliding the [b]MoveMe[/b] slider.[br][br]You can modify any of these parameters (functions, limits) at any time. [br][br][color=#1e84cc][b]To explore in augmented reality, see the directions below this applet. [/b][/color]
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device. [br][br]2) Go to the MENU (horizontal bars) in the upper left corner. Select OPEN. [br] In the Search GeoGebra Resources input box, type [b]rfjatd4f[/b][br] (Note this is the resource ID = last 8 digits of the URL for this resource.)[br][br]3) In the resource that uploads, zoom in/out if needed.[br] You can alter the functions upper(x), lower(x). [br] You can also move sliders a, b, MoveMe, and Filling.[br] (Be sure to leave everything else alone.)
Modifiable Solid: Rectangular Cross Sections Parallel to yAxis
HOW TO USE:
Enter any [b]UPPER FUNCTION[/b] (labeled [b]upper(x)[/b]).[br]Enter any [b]LOWER FUNCTION[/b] (labeled [b]lower(x)[/b]).[br]Enter your [b]lower and upper limits of integration[/b] ([b]"a"[/b] and [b]"b"[/b], respectively).[br][br]Slide the [b]FILLING[/b] slider all the way to the left. [br]For this solid, note how cross sections parallel to the yAxis are [color=#ff7700][b]rectangles[/b][/color].[br]One side of this [color=#ff7700][b]rectangle[/b][/color] (the base) has length = upper(x) - lower(x). [br]The other side (height) of this [b][color=#ff7700]rectangle[/color][/b] has length = w * (upper(x) - lower(x)). [br][br]Thus, the height of any rectangular cross section = w * its base length. [br]You can alter the value of [b]w[/b] using the (vertical) [b]w slider[/b]. [br][br][br]You can move this [color=#ff7700][b]rectangle[/b][/color] by sliding the [b]MoveMe[/b] slider OR by dragging the [b]WHITE POINT [/b]along the xAxis. [br][br]You can modify any of these parameters (functions, limits) at any time. [br][br][color=#1e84cc][b]To explore in augmented reality, see the directions below this applet. [/b][/color]
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device. [br][br]2) Go to the [b]MENU [/b](horizontal bars) in the upper left corner. Select [b]OPEN[/b]. [br] In the Search GeoGebra Resources input box, type [b]mhnnfubg[/b][br] (Note this is the resource ID = last 8 digits of the URL for this resource.)[br][br]3) In the resource that uploads, zoom in/out if needed.[br] You can alter the functions [b]upper(x), lower(x).[/b] [br] You can also move sliders [b]a, b, w, MoveMe, and Filling[/b].[br] (Be sure to leave everything else alone.)
Building Surfaces with Cross Sections and Function Modeling
Note the dimensions of the Toblerone shown below.
The Toblerone candy bar shown is an example prism. By definition, what exactly makes a 3D solid a [i]prism[/i]? Describe.
The solid below has equilateral triangle cross sections parallel to the yAxis. Given the dimensions of the Toblerone shown above, construct an accurate model of it below.
Create a solid whose base is a circle with radius 8 and has equilateral triangle sections parallel to the yAxis. (Cross sections shown here are equilateral triangles.)
This solid has 45-45-90 triangle cross sections parallel to the yAxis (leg lies in base). Create a solid with a base that is similar to these cross sections so that its (base) area = 50.
Which of the following solids has a volume that can be expressed as a [b]single[/b] definite integral and not the sum of one or more definite integrals? Check all that apply.