\[ \iint _R \d A \]
where \(R\) is the square with
corners \((0,0)\), \((1,1)\), \((2,0)\) and \((1,-1)\). Without computing any antiderivatives, evaluate
Without computing any antiderivatives, evaluate
\[ \iint _R xy \d A \]
where \(R\) is the square with
corners \((0,0)\), \((1,1)\), \((2,0)\) and \((1,-1)\). Evaluate
\[ \iint _R xy^2 \d A \]
where \(R\) is the square with corners \((0,0)\), \((1,1)\), \((2,0)\) and \((1,-1)\). Consider \(\int _{0}^{1}\int _{0}^{x}\int _{0}^{\sqrt {x^2+y^2}}\d z\d y \d x\). Sketch the region we are integrating over.
Let \(R\) be the region defined to be the interior of the cylinder \(x^2+y^2=1\) between \(z=0\) and \(z=5\). Sketch
this region and evaluate \(\iiint _R x^2 \d V\).
Let \(R\) be the region in the first octant, inside \(x^2+y^2-2x=0\) and under \(x^2+y^2+z^2=4\). Sketch this region and
evaluate \(\iiint _R yz \d V\).
Evaluate \(\iiint _R \sqrt {x^2+y^2} \d V\) over the interior of \(x^2+y^2+z^2=4\).