Calculating Mass for a Cylinder in the First Octant

[cp255]

If the mass per unit area of a surface is given by ρ=xy, find the mass if S is the part of the cylinder x²+z²=25 which is in the first octant and contained within the cylinder x²+y²=16.So here was my attempt.

I parametrized the curve.
x²+z²=25
r(u, v) = <5cos(u), v, 5sin(u)>

I then plugged into the bounds
x²+y²=16.
25cos²(u) + v² = 16
v = sqrt(16 - 25cos²(u))

Next I took the cross product of r_u X r_v. Its magnitude is a constant 5.

Now I solved the integral with bounds 0 < u < Pi/2 and 0 < v < sqrt(16 - 25cos²(u))
∫∫25 v cos(u) du dv

This returns the result of -25/3 but since we are looking for a mass I submitted the answer of positive 25/3 and this is wrong. I checked the integration on my calculator and it gets the same result.

 

[LCKurtz]

If the mass per unit area of a surface is given by ρ=xy, find the mass if S is the part of the cylinder x²+z²=25 which is in the first octant and contained within the cylinder x²+y²=16.


So here was my attempt.

I parametrized the curve.
x²+z²=25
r(u, v) = <5cos(u), v, 5sin(u)>

I then plugged into the bounds
x²+y²=16.
25cos²(u) + v² = 16
v = sqrt(16 - 25cos²(u))

Next I took the cross product of r_u X r_v. Its magnitude is a constant 5.

Now I solved the integral with bounds 0 < u < Pi/2 and 0 < v < sqrt(16 - 25cos²(u))
∫∫25 v cos(u) du dv

This returns the result of -25/3 but since we are looking for a mass I submitted the answer of positive 25/3 and this is wrong. I checked the integration on my calculator and it gets the same result.

##u## doesn't go from ##0## to ##\pi/2##. Look at a picture showing what angle ##u## represents.

 

[cp255]

I redid my bounds and now I have arccos(sqrt(16-v^2)/5) < u < pi/2 and 0 < v < 4 which results in the answer of 22/15 which is still wrong. I fell like I found the wrong value for u.

 

[LCKurtz]

I get 22/3 both that way and working the integral in reverse order so I think you have a mistake somewhere. But haven't you left out multiplying by the multiplier ##|r_u\times r_v|##? That would give 110/3 by my calculations. Is that the given answer?