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Volumes of Revolution Word Problem

  1. Mar 19, 2008 #1
    1. The problem statement, all variables and given/known data

    Assume that the Earth is a sphere with circumference of 24,900 miles.
    a. Find the volume of the Earth north of latitude 45 degrees. (hint: integrate with respect to y)
    b. Find the volume of the Earth between the equator and latitude 45

    2. Relevant equations

    circle: x^2 + y^2 = r^2

    3. The attempt at a solution

    so far, I have just been working on A. I took a cross section of the sphere from latitude 45 and up and drew it on a graph. I realized that if i revolved it around the y-axis that it would form the shape I need, a dome.

    I found the radius using the circumference and set up my integral.
    i have: pi * int(124500pi - y^2 dy after simplifying.

    I think i'm all set to integrate and find the answer, but I cant figure out what to use for the upper and lower bounds. I thought i might try and use sin(45) or cos(45) or tan(45), in some way, but I cant really wrap my head around the problem from this point forward.
  2. jcsd
  3. Mar 19, 2008 #2


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    Homework Helper

    … avoid using huge numbers … !

    Hi xcgirl! :smile:

    Hint: with big numbers like this, just put the radius = r throughout the calculation, and then put the number for r in at the end - you're much less likely to make a mistake (like forgetting to square something!) - and you won't have five-digit limits for the integral sign!

    Yes, your approach seems fine.

    Integrate over y, from y = r/√2 to r. :smile:

    (If in doubt as to whether it's sin or tan, draw a diagram!)
  4. Mar 19, 2008 #3
    when i do this, i am getting very large numbers, larger even then the actual volume of the earth. I'm not sure whats going on.
  5. Mar 19, 2008 #4
    The integral you might be using, which seems to be

    [tex]V = \pi\int_{\frac{r}{\sqrt{2}}}^r (124500\pi - y^2)dy[/tex]

    doesn't look right. You may have simplified wrong; from where did you get [itex]124500\pi[/itex]?
  6. Mar 20, 2008 #5


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    … let's see …

    Hi xcgirl! :smile:

    Show us the integral you used, before putting any numbers in (ie just using r), so we can see what is going wrong. :smile:
  7. Mar 18, 2010 #6
    http://www4a.wolframalpha.com/Calculate/MSP/MSP103319a0269e42g61e9e00000ga3250hih542d42?MSPStoreType=image/gif&s=35&w=114&h=48 [Broken]
    This is the integral I used and I get 1.6376*10^12 miles cubed which doesn't seem right.
    Last edited by a moderator: May 4, 2017
  8. Mar 19, 2010 #7


    Staff: Mentor

    If the earth were a cube 8000 miles on each side, its volume would be (8,000)3 mi3 = 512 x 109 mi3 = 5.12 x 1011 mi3. Being roughly spherical, the earth would fit inside such a box, so its volume would be less than this. That makes the value too big by maybe two orders of magnitude, since you're calculating the volume above 45 degrees N.
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