If the mass per unit area of a surface is given by p=xy , find the mass (double integral over S) xydS if S is the part of the cylinder y^2+z^2=16 which is in the first octant and contained within the cylinder x^2+y^2=1.
Explanation / Answer
S: x²+z²=5² ==> 2x+2z*( ?z/?x)=0, ?z/?x=-x/z; ?z/?y=0 dS=v[1+(?z/?x)²+(?z/?y)²] dx dy=v(1+x²/z²) dx dy= v((x²+z²)/z²) dx dy=5/|z| dx dy=5/v(25-x²) dx dy Let D be the orthogonal projection in the xOy plane (z=0) of the part of S which is in the first octant and contained within the cylinder x^2+y^2=16 ==> D is bounded by the circle x²+y²=4² and x and y axes ==> D: 0