Studying? (average value, work)

[thename1000]

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 don't 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.

 

[lippyka]

a) To find the average value of the function, we need to calculate the integral of the function over the given interval. The integral is given by:∫f(x)dx from 1 to 10 = ∫3/(1+x)^2 dx from 1 to 10 = [3 ln (1+x)]/[(1+x)^2] from 1 to 10Therefore, the average value of the function on the interval x=1 to x=10 is:Average = [3 ln (11) - 3 ln (2)]/[(11)^2 - (2)^2]b) The work required to pull the cable to the top can be calculated as follows:Work = Force x Distance = 50 lb x 40 ft = 2000 ft-lb c) The hydrostatic force against the gate can be calculated using the equation:Force = ρghA where ρ is the density of water (9800 N/m^3), g is the acceleration due to gravity (9.8 m/s^2), h is the height of the dam (unknown), and A is the area of the gate (πr2). Therefore, the hydrostatic force against the gate is:Force = 9800 x 9.8 x h x (πr2) d) The volume of the solid generated by revolving the region bounded by y=3sqrt(x) and the x-axis from x=0 to x=5 can be found using the equation:Volume = ∫πy2dx from 0 to 5 = π ∫x(3sqrt(x))^2dx from 0 to 5 = π/4 ∫(9x^(5/2))dx from 0 to 5 = [9πx^(7/2)]/7 from 0 to 5 = [9π(125^(7/2)) - 9π(0^(7/2))]/7 = 27