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Studying? (average value, work)

  1. Oct 7, 2007 #1
    I'm studying for a test and it would be great if i could get step by step how to do this problem:

    a.) Find the average value of the function on the interval x=1 to x=10 for f(x)=3/(1+x)^2

    b.) A uniform cable hanging over the edge of a tall building is 40 ft long and weighs 50 lb. How much work is required to pull the cable to the top?

    c.) A vertical dam has a semicircular gate as shown in the figure below(http://img223.imageshack.us/my.php?image=newwt5.jpg), The density of water is 9800 Newtons per cubic meter. Find the hydrostatic force against the gate.

    d.) Find the volume of the solid generated by revolving about the y-axis the region bounded by the x-axis and y=3x-x^3 from x=0 to x=5 (http://img223.imageshack.us/my.php?image=newwt5.jpg)

    e.) For the lamina of density P formed by the region bounded by y=3sqrt(x) {NOT 3 times sqrt x} and the x-axis from x=0 to x=8, find the y coordinate of the centroid. (http://img223.imageshack.us/my.php?image=newwt5.jpg)


    I dont need the final answers, but enough so that I can follow what your doing. (If you only know how to do one of these that'd be fine!) Thanks.
     
  2. jcsd
  3. Oct 7, 2007 #2
    a.)
    The average value of a function between intervals [tex][a,b][/tex] is defined as:
    [tex]\frac{1}{b-a} \int_a^b f(x)\,dx [/tex]

    and just in case:
    [tex]\int kx^n\,{\rm d}x = k\frac{x^{n+1}}{n+1} + C[/tex] and
    [tex]\frac{3}{(x+1)^2} = 3(x+1)^{-2}[/tex].
     
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