How Do You Solve These Calculus Problems?

[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.

 

[atqamar]

a.)
The average value of a function between intervals [a,b] is defined as:
\frac{1}{b-a} \int_a^b f(x)\,dx

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