Use Stokes' Theorem to Evaluate $\int _ { C } ( x z + 2 y ) d x + \left( x ^ { 2 } + 5 y z \right) d y + \left( y ^ { 2 } + z \right) d z$

Short Answer

Use Stokes' Theorem to evaluate $\int _ { C } ( x z + 2 y ) d x + \left( x ^ { 2 } + 5 y z \right) d y + \left( y ^ { 2 } + z \right) d z$ , where C is the curve of intersection of the paraboloid $z = x ^ { 2 } + 5 y ^ { 2 }$ and the cylinder $( x - 4 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 16$ , oriented counterclockwise as viewed from above.