Volume of Revolution with Applet

Investigate the Rotation of a Line about x-axis

use this applet to enter y = 3 , y = x+3 and [math]y=\sqrt{x+2}+1[/math] .from a= -3 to b= 3. [br]Use spin Slider to rotate the lines or curves about x.axis.

Q1.Investigate the Rotation using Spin slider, Enter y = 3 and Set a= -3 and b= 3.

Spin it up to 360 degrees. [br]Which Solid shape is formed?[br]What is the radius and height of that shape?

Q1 a. Can you work out the volume of the Shape formed

What would be your procedure for Volume of the shape?

Q2 a, Enter y = x+3, from a= -3 to b= 3, Spin it and move slider w

You may Untick the unhide rotation and disc boxes to see them clearly. Describe the Volume of the Shape formed?

Q2b: y= x+3

When you rotate y = x+3 and you get the shape of the solid formed. Do you know the formula of volume of that Shape?

Q3. Untick : Unhide the Rotation

The Red disc ( Circular shaped element), If you move slider w, what is changing with w?

Q4. The Slider w is changing the radius of circular disc.

The Radius at different positions is actually given by the value of the function (y= f(x)) at different x values.[br]So Remember radius of the Disc at 'x= w' would be f(w). if the width of the disc is 'dx' then what is the Volume of the Disc?

Q5, Select the Correct option

as w is the variable, we can say the radius of the disc = f(x) ( at any value of x). and the width is dx. The Volume of the disc (Cylinder) would be

[math]Volume=\pi\left(w\right)^2\left(width\right)[/math] [math]Volume=\pi\left(R\right)^2\left(dx\right)[/math] [math]Volume=\pi\left(R\right)^2\left(width\right)=\pi\left(f\left(x\right)\right)^2\left(dx\right)[/math]

Q6 if there are n- number of discs making a solid shape when you rotate a line or curve about x-axis from a to b

Then The Total Volume should be the sum of the volume of all discs and it is given by

[math]\sum^{x=b}_{x=a}\left[\pi\left(f\left(x\right)\right)^2\left(dx\right)\right][/math] [math]Volume=\sum\left[\pi\left(f\left(x\right)\right)^2\left(dx\right)\right][/math] [math]\sum^{x=b}_{x=a}\left[\pi\left(f\left(x\right)\right)^2\left(dx\right)\right]=\pi\int_a^b\left[\left(f\left(x\right)\right)^2\left(dx\right)\right][/math]