Making sure Volume

[layd33foxx]

I just want to make sure that I start this equation correct.

V=2π ∫₀^.07 (y)(1)dy + 2π ∫_^.15.07 (y)((1/y)-1)dy

not sure if it's .15 or .1??

 

[tkhunny]

Several issues.

Most important, the problem is ill defined. Is it REALLY on [7,8] or are we just to BELIEVE that is what it is. It should be stated explicitly.

Next, you have a curve, y = 1/x, why are you using odd decimals rather than exact values? (8,1/8), (7,1/7). No need for 0.150 or 0.125.

Why are you using an integral for the volume of a right circular cyllinder? Do it once to prove it works, then never do it again.

 

[layd33foxx]

using an integral because I have to to answer it? I mean what other way is there this?V = πr2h Will it get to the right answer?

 

[tkhunny]

Okay, that was one of my objections. Please understand that the intergral expression will result in the well-known formula. It's UNIQUE! Why would it be different?

 

[layd33foxx]

I don't know haven't thought about it that way..

 

[soroban]

Hello, layd33foxx!

First of all, I'm stunned that a textbook would use decimal approximation . . . awful!
Second, why would anyone want to do this with "shells"?

I just want to make sure that I start this equation correct.

\(\displaystyle \displaystyle V \:=\: 2\pi \int^{0.7}_0(y)(1)\,dy + 2\pi \int^{0.15}_{0.07}(y)\left(\tfrac{1}{y}-1\right)\,dy\)

I don't understand your limits.

      |
  1/7 +           *
      |           |:*..
  1/8 +           *:-:-:*
      |           |:::::|
      |           |:::::|
      |           |:::::|
      |           |:::::|
  - - + - - - - - + - - + - -
      |           7     8

Revolved about the x-axis, we have a "wheel".

The center is a flat cylinder with radius 1/8 and height 1.
Its volume is: .\(\displaystyle V_1 \:=\:\pi (\frac{1}{8})^2(1) \:=\:\frac{\pi}{64}\)
It can be calculated with: .\(\displaystyle \displaystyle V_1 \:=\:2\pi\int^{\frac{1}{8}}_0 (y)(1)\,dy \:=\:\frac{\pi}{64}\)

The outer rim has: .\(\displaystyle \displaystyle V_2 \:=\:2\pi\int^{\frac{1}{7}}_{\frac{1}{8}} y(y-7)\,dy \:=\:\frac{\pi}{448}\)

The total volume is: .\(\displaystyle V \:=\:\dfrac{\pi}{64} +\dfrac{\pi}{448} \:=\:\dfrac{\pi}{56}\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The answer can be verified by the "disk" method.

\(\displaystyle \displaystyle V \;=\;\pi \int^8_7 \left(\frac{1}{x}\right)^2\,dx \;=\;\pi\int^8_7 x^{-2}\,dx \;=\;\pi\bigg[-\frac{1}{x}\,\bigg]^8_7\)

. . . . . \(\displaystyle =\;\pi\bigg[\left(-\dfrac{1}{8}\right) - \left(-\dfrac{1}{7}\right)\bigg] \;=\;\dfrac{\pi}{56} \)

 

[tkhunny]

I don't know haven't thought about it that way..

This is a tough warning sign. Perhaps it is time for you to figure out that you are not just messing with little symbols and you are actually doing something.

 

[layd33foxx]

I'm not messing around with little symbols it's a real question haha ...I'll do this problem real soon and I'll get back to check my answer.