91Ó°ÊÓ

\(11-14\) Sketch the solid described by the given inequalities. $$ \rho \leqslant 1, \quad 0 \leqslant \phi \leqslant \pi / 6, \quad 0 \leqslant \theta \leqslant \pi $$

\(11-12\) Sketch the solid described by the given inequalities. $$ r^{2} \leqslant z \leqslant 8-r^{2} $$

A lamina occupies the part of the disk \(x^{2}+y^{2} \leqslant 1\) in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the \(x\) -axis.

Evaluate the triple integral. $$ \begin{array}{l}{\iiint_{E} \sin y d V, \text { where } E \text { lies below the plane } z=x \text { and above }} \\ {\text { the triangular region with vertices }(0,0,0),(\pi, 0,0), \text { and }} \\ {(0, \pi, 0)}\end{array} $$

Evaluate the given integral by changing to polar coordinates. \(\iint_{D} \cos \sqrt{x^{2}+y^{2}} d A,\) where \(D\) is the disk with center the origin and radius 2

A region \(R\) in the \(x y\) -plane is given. Find equations for a transformation \(T\) that maps a rectangular region \(S\) in the \(u v\) -plane onto \(R,\) where the sides of \(S\) are parallel to the \(u\) - and \(v\) -axes. $$ \begin{array}{l}{R \text { is the parallelogram with vertices }(0,0),(4,3),(2,4)} \\ {(-2,1)}\end{array} $$

Sketch the solid described by the given inequalities. $$ 1 \leqslant \rho \leqslant 2, \quad \pi / 2 \leqslant \phi \leqslant \pi $$

A region \(R\) in the \(x y\) -plane is given. Find equations for a transformation \(T\) that maps a rectangular region \(S\) in the \(u v\) -plane onto \(R,\) where the sides of \(S\) are parallel to the \(u\) - and \(v\) -axes. $$ \begin{array}{l}{R \text { lies between the circles } x^{2}+y^{2}=1 \text { and } x^{2}+y^{2}=2 \text { in }} \\ {\text { the first quadrant }}\end{array} $$

\(13-14\) Find \(\int_{0}^{2} f(x, y) d x\) and \(\int_{0}^{3} f(x, y) d y\) $$ f(x, y)=x+3 x^{2} y^{2} $$

Evaluate the given integral by changing to polar coordinates. \(\iint_{R} \arctan (y / x) d A\) where \(R=\left\\{(x, y) | 1 \leqslant x^{2}+y^{2} \leqslant 4,0 \leqslant y \leqslant x\right\\}\)