A tale of eight integrals

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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:

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

$\oiint _B \vec {F}\dotp \uvec {n}\d S$

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