## 5 calc problems that i need checked!

Hey I am new here, and i am desperately in need of help.

1. Find the Hydrostatic Force on the plate. Image - TinyPic - Free Image Hosting, Photo Sharing & Video Hosting

Force = integral from a-b of density x height x length x width
I got F = integ from 0-3 (62.5)(4-y)((4/3)y)dy

I got the length by taking two points (4,3) and (0,0) and finding the equation y-3 =3/4x (x-4) and thus i got y=(3/4)x or x=(4/3)y

The answer i got was 750. Let me know if this is correct.

2. Find the center of mass of a lamina: Image - TinyPic - Free Image Hosting, Photo Sharing & Video Hosting.
I only need the x-bar for this one.. not the y bar.

since the top if a semi circle i got x^2+y^2=1 and the bottom eq is y =x
so isolating the y i get y= root(1-x^2)

x-bar eq: (1/Area) x integ(a to b) xf(x)dx

I got as far as [integ from 0-1 x(root(1-x^2) -x)] / [integ from 0-1 root(1-x^2) -x]

Then im stuck.

3. Find the length of the curve: y = ln(cosx), 0 lessthanorequalto x lessthanorequalto pi/3

This one I got ln|2+root(3)| through standard integration and 1 identity using the formula: integral of root(1+(f'(x)^2))

4. Find the area of the surface obtained by rotating the curve about y-axis: x^2+y^2=r^2, 0 lessthanorequalto y lessthanorequalto r/2

This one i just isolated the x and got x=root(r^2-y^2) and found the derivative to be (1/2)(r^2-y^2)^-1/2 then i got stuck.

5.Find the exact coordinates of the centroid. Find the x bar and y bar.
the two formulas: y=x and y=rootx.

i used the x-bar formula of (1/A)integ(a-b)x(f(x)-g(x))dx and got 6/15
i used the y-bar formula of (1/A)integ(a-b)(1/2)(f(x)^2 - g(x)^2)dx and got 1/2.