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
$$\frac{3}{(x+1)^2} = 3(x+1)^{-2}$$.

 

[tacman]

a.) To find the average value of a function on an interval, we use the formula: (1/(b-a)) * integral from a to b of f(x) dx. In this case, a=1, b=10, and f(x)=3/(1+x)^2. So, the average value would be: (1/(10-1)) * integral from 1 to 10 of (3/(1+x)^2) dx = (1/9) * (-(1/1+x)) from 1 to 10 = (1/9) * (-(1/11) + (1/2)) = 0.055.

b.) To find the work required, we use the formula: W = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, m=50 lb, g=32 ft/s^2, and h=40 ft. So, the work required would be: (50)(32)(40) = 64,000 ft-lb.

c.) To find the hydrostatic force, we use the formula: F = (density)(gravity)(volume). In this case, the density is given as 9800 Newtons per cubic meter, and the volume would be half of a cylinder. So, the volume would be: (1/2)(pi)(radius)^2(height), where the radius is 4 meters and the height is 8 meters. Substituting these values into the formula, we get: F = (9800)(9.8)(1/2)(pi)(4)^2(8) = 2,457,600 Newtons.

d.) To find the volume of the solid generated, we use the formula: V = pi * integral from a to b of (f(x))^2 dx. In this case, a=0, b=5, and f(x)=3x-x^3. So, the volume would be: pi * integral from 0 to 5 of ((3x-x^3)^2) dx = pi * integral from 0 to 5 of (9x^2 - 6x^4 + x^6) dx = pi * (3x^3 - (6/5)x^5 + (1/7)x^7) from 0 to 5 = pi * (375/