Area b/w Curves and Disk/Shell/Washer Method

[LongTermStudent]

I'm struggling with a couple of problems here. I'm not sure how to write fractions and integrals properly on this forum, so I will do my best to make it legible. I'll write integrations as such (int,ub,lb) and that will hopefully alleviate confusion (for me anyway).

• Use the shell method to set up the integral that represents the volume of the solid formed by revolving the region bounded by the graphs of y = 1/x and 2x + 2y = 5 about the line y = 1/2
  
  I rearrange these equations so that x = 1/y and x = 5/2 - y because it is a horizontal axis of rotation. I then come up with
  2pi [(int,2,1/2) 5/(2y) - 1]dy
  [/*:m:2o3oooda]
• Find the volume of the solid formed by revolving the region bounded by the graphs of y = x² and y = 4 about the x-axis
  
  This is a vertical axis of revolution, so I set up the integral as such: 2pi[(int,0,2)x³]dx
  
  Working that out I come up with 8pi
  [/*:m:2o3oooda]

Last question...

• Write the definite integral that represents the arc length of one period of the curve y = sin2x
  
  Since y = sin2x, y' = 2cos2x. This means the arc length
  
  s = \(\displaystyle \L\\2\int_{0}^{{\pi}/2}sqrt(1+(2cos2x)^2)dx\)
  
  which I simplified to
  
  s = \(\displaystyle \L\\2\int_{0}^{{\pi}/2}sqrt(1+4cos^2(2x))dx\)
  
  My reasoning for multiplying the integral by 2 is so that I account for the section of the arc above the x-axis and the section of the arc below the x-axis.[/*:m:2o3oooda]

I hope this isn't too difficult to read, and I appreciate any and all advice given.

 

[galactus]

We can do this with shells. Do it with washers for a check.


Shells:

\(\displaystyle \L\\2{\pi}\int_{1/2}^{2}(y-\frac{1}{2})(\frac{5}{2}-y-\frac{1}{y})dy\)


Washers:

\(\displaystyle \L\\{\pi}\int_{\frac{1}{2}}^{2}(\frac{1}{2}-(\frac{5}{2}-x))^{2}-(\frac{1}{2}-\frac{1}{x})^{2}dx\)


Same result with both methods.


Here's what it looks like:

If you wish to display in a nice format(like I just used), learn a little LaTex.
Click on quote to see the code I used to type what I posted. No software needed. Just the right code.

 

[LongTermStudent]

Thanks, I struggle with finding volumes when rotating around something besides the x or y-axis. Since nobody has said otherwise, is it safe to assume my work is ok on the other problems? I know I'm supposed to limit my questions, so I hope asking three wasn't too much for my first time.

On a side note, I'm having no luck with the LaTex, is there a tag I'm supposed to use in conjunction with the \L\\... coding? All I get is
\(\displaystyle \L\\2{\pi}\int_{1/2}^{2}(y-\frac{1}{2})(\frac{5}{2}-y-\frac{1}{y})dy\)
if I type what I see when I mouse-over your quote. Also, that graphic was very helpful in visualizing the problem. What kind of software did you use to generate that?

I fix.

 

[galactus]

You must enclose in

[tex][/tex]

. You should've seen that with my code. As for the solid of revolution diagram........that's Maple 10.


As for the second one, it's a horizontal axis. The x-axis is horizontal.

Shells:

\(\displaystyle \L\\2{\pi}\int_{0}^{4}y^{\frac{3}{2}}dy\)


Washers:

\(\displaystyle \L\\{\pi}\int_{0}^{2}(16-x^{4})dx\)