Simpsons rule for volume (1 Viewer)

[akuchan]

X_X" dunno what im doing wrong but heres the question

The curve y = 2^x is rotated about the x-axis between x = 1 and x = 2. Use Simpsons rule with 3 function values ..find volume... 3 significant figures

the answer is 27.2 units cubed
while i keep gettin 17.2 units cubed, well its getting late and im getting dizzy =/ maybe thats why lol. but yeh.. its a bit hard to type up the answer without the integration symbols and stuff, but any help is appreciated

 

[PC]

Use the function y = 2^x to find the function values.
These are all nice points above the x-axis, so they are effectively radii of circles, once the curve is rotated around the x-axis.

So, now you can find the area of these three circles since you know the radii!

Area circle 1 = πr² = π x (2¹)² = 4π
Area circle 2 = πr² = π x (2^1.5)² = 8π
Area circle 3 = πr² = π x (2²)² = 16π

Rather than use Simpson's Rule to find the area, use it to find the volume.

Volume = h/3 x (A₁ + 4A₂ + A₃)
= 0.5/3 x (4π + 4 x 8π + 16n)
= 0.5/3 x 52π
= 27.227136333
= 27.2 cubic units

 

[akuchan]

ohhh nice, thanks
could someone also do it using the proper formula too
pi [integrate]b/a y^2 dx

its not actually b/a.. but i dunno how else to type it lol, witht hat formula, would it only work with cylinders, spheres and cones since thats where its devised from?

also back to the simpsons rule thing, do u always find the 'area of the circles' at the function values then sub it in the formula to get the volume, before i was just subbing the y^2 into the formula and then multiply the whole thing by pi =/

 

[iflybees]

Okay
(So very late, but this may help anyone else who's interested)

Cause it's volume you have to integrate it first to get the y^2 part

Using the form
V = π b∫a y^2 dx
(imagine the b is on the top and a is on the bottom)

You need to make the y=2^x into y^2 = 2^2x

And from there, do your table of values with 3 function values.

x 1 1.5 2
y 4 8 16

Then, using those- sub it into the simpson's rule.

where h= 2-1/2
= 0.5

therefore
V = 0.5/3 (4+16+4x8)
= 26/3
multiply by π to get the volume
= 27.2 u^3

I hope that helps!