Vector Calculus-Painting a fence

[Daniel_Feldman]

John needs to paint a fence that encloses his field. The fence is circular, and its base (in yards) is decribed by the parametric equations x=18cos(theta) and y=18sin(theta). The fence's height (in feet) is given by h(x,y)=12+(2x-y)/6. Assuming that one gallon of paint covers 300 square feet of fence, how many gallons of paint will john need to complete the project?


I'm not sure how to go about this one. I think it involves a line integral of some sort, but am not sure.

 

[Daniel_Feldman]

You need the extra r too right, since it's in polar coordinates?

 

[galactus]

The surface area of a cylinder is given by \(\displaystyle 2{\pi}rh\)

\(\displaystyle \L\\2{\pi}(18)\left(\frac{2(18cos(t))-18sin(t)}{6}\right)=108{\pi}(2cos(t)-sin(t))\)


\(\displaystyle \L\\4\int_{0}^{\frac{\pi}{2}}\left[108{\pi}(2cos(t)-sin(t))\right]dt\)


=\(\displaystyle \L\\432{\pi}\)

\(\displaystyle \L\\\frac{432{\pi}}{300}=4.52 \;\ gallons\)

That seems plausible. Same as before.