Work in groups of 3–4, writing your answers on a separate sheet of paper.

Let \(T\) be the region bounded by the triangle whose vertices are:

\[ (1,2)\quad (3,4)\quad (4,1) \]

Also let \(P\) be the polygonal path from \((1,2)\) to \((3,4)\) to \((4,1)\) (the triangle above, missing its bottom side). Compute the following integrals:

\(\iint _T \d A\)
\(\oint _{\partial T} \left ((2x+3y)\d x + (3x+3y^2)\d y\right )\)
\(\int _P \left ((2x+3y)\d x + (3x+3y^2)\d y\right )\)
\(\oint _{\partial T} \left ((3x^2+3y)\d x + (x+3y)\d y\right )\)
\(\int _P \left ((3x^2+3y)\d x + (x+3y)\d y\right )\)

Let:

  • \(B\) be surface of the ball of radius \(9\) centered at the point \((-1,2,3)\), oriented outward,
  • \(H\) be the upper-half of \(B\), after it is cut by the plane \(x+2y-z =0\), oriented upward in the positive \(z\)-direction,
  • \(\vec {F} = \vector {3x-2y,z-y,x+2z}\).

Compute the following integrals:

\(\oiint _B (\curl \vec {F})\dotp \uvec {n}\d S\)
\(\oiint _B \vec {F}\dotp \uvec {n}\d S\)
\(\iint _H (\curl \vec {F})\dotp \uvec {n}\d S\)