6.6 Surface integrals  (Page 15/27)

[T] Evaluate ${\displaystyle {\iint }_S\left(z+y\right)dS},$ where S is the part of the graph of $z=\sqrt{1-x^2}$ in the first octant between the xz -plane and plane $y=3.$

Evaluate ${\displaystyle {\iint }_{S}xyzd\text{S}}$ if S is the part of plane $z=x+y$ that lies over the triangular region in the xy -plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).

Find the mass of a lamina of density $\xi (x,y,z)=z$ in the shape of hemisphere $z={\left(a^2-x^2-y^2\right)}^{1\text{/}2}.$

Compute ${\displaystyle \int {\displaystyle {\int }_S\text{F}·\text{N}dS,}}$ where $\text{F}(x,y,z)=x\text{i}-5y\text{j}+4z\text{k}$ and N is an outward normal vector S , where S is the union of two squares $S_1:x=0\text{,}0\le y\le 1\text{,}0\le z\le 1$ and $S_2:z=1\text{,}0\le x\le 1\text{,}0\le y\le 1.$

Compute ${\displaystyle \int {\displaystyle {\int }_S\text{F}·\text{N}dS,}}$ where $\text{F}(x,y,z)=xy\text{i}+z\text{j}+(x+y)\text{k}$ and N is an outward normal vector S , where S is the triangular region cut off from plane $x+y+z=1$ by the positive coordinate axes.

Compute ${\displaystyle \int {\displaystyle {\int }_S\text{F}·\text{N}dS,}}$ where $\text{F}\left(x,y,z\right)=2yz\text{i}+\left({\text{tan}}^{\mathrm{-1}}xz\right)\text{j}+e^{xy}\text{k}$ and N is an outward normal vector S , where S is the surface of sphere $x^2+y^2+z^2=1.$

Compute ${\displaystyle \int {\displaystyle {\int }_S\text{F}·\text{N}dS,}}$ where $\text{F}(x,y,z)=xyz\text{i}+xyz\text{j}+xyz\text{k}$ and N is an outward normal vector S , where S is the surface of the five faces of the unit cube $0\le x\le 1\text{,}0\le y\le 1\text{,}0\le z\le 1$ missing $z=0.$

${\displaystyle {\iint }_{S}x{y}^2{z}^{3}d\text{S}};$ S is the first-octant portion of plane $2x+3y+4z=12.$

${\displaystyle {\iint }_S\left(x^2-2y+z\right)d\text{S;}}$ S is the portion of the graph of $4x+y=8$ bounded by the coordinate planes and plane $z=6.$

${\displaystyle {\iint }_{S}x{y}^2{z}^{3}d\text{S}};$ S is the first-octant portion of plane $2x+3y+4z=12.$

${\displaystyle {\iint }_S\left(x^2-2y+z\right)d\text{S;}}$ S is the portion of the graph of $4x+y=8$ bounded by the coordinate planes and plane $z=6.$

Evaluate surface integral ${\displaystyle {\iint }_{S}yzd\text{S}},$ where S is the first-octant part of plane $x+y+z=\lambda ,$ where $\lambda$ is a positive constant.

Evaluate surface integral ${\displaystyle {\iint }_S\left(x^{2}z+y^{2}z\right)d\text{S}},$ where S is hemisphere $x^2+y^2+z^2=a^2,z\ge 0.$

Evaluate surface integral ${\displaystyle {\iint }_{S}zd\text{A}},$ where S is surface $z=\sqrt{x^2+y^2},0\le z\le 2.$

Evaluate surface integral ${\displaystyle {\iint }_S{x}^{2}yzdS},$ where S is the part of plane $z=1+2x+3y$ that lies above rectangle $0\le x\le 3\text{and}0\le y\le 2.$

Evaluate surface integral ${\displaystyle {\iint }_{S}yzdS,}$ where S is plane $x+y+z=1$ that lies in the first octant.

Evaluate surface integral ${\displaystyle {\iint }_{S}yzd\text{S}},$ where S is the part of plane $z=y+3$ that lies inside cylinder $x^2+y^2=1.$

${\displaystyle {\iint }_S\sqrt{x^2+y^2+z^2}dS},$ where S is surface $x^2+y^2+z^2=4,z\ge 0$

${\displaystyle {\iint }_S(x\text{i}+y\text{j})·d\text{S}},$ where S is surface $x^2+y^2=4,1\le z\le 3,$ oriented with unit normal vectors pointing outward

${\displaystyle {\iint }_S(z\text{k})·d\text{S}},$ where S is disc $x^2+y^2\le 9$ on plane $z=4,$ oriented with unit normal vectors pointing upward

A lamina has the shape of a portion of sphere $x^2+y^2+z^2=a^2$ that lies within cone $z=\sqrt{x^2+y^2}.$ Let S be the spherical shell centered at the origin with radius a , and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z -axis. Determine the mass of the lamina if $\delta (x,y,z)=x^2{y}^{2}z.$

A lamina has the shape of a portion of sphere $x^2+y^2+z^2=a^2$ that lies within cone $z=\sqrt{x^2+y^2}.$ Let S be the spherical shell centered at the origin with radius a , and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z -axis. Suppose the vertex angle of the cone is ${\varphi }_0,\text{with}0\le {\varphi }_0<\frac{\pi }{2}.$ Determine the mass of that portion of the shape enclosed in the intersection of S and C . Assume $\delta (x,y,z)=x^2{y}^{2}z.$

A paper cup has the shape of an inverted right circular cone of height 6 in. and radius of top 3 in. If the cup is full of water weighing $62.5\text{lb}\text{/}{\text{ft}}^3,$ find the total force exerted by the water on the inside surface of the cup.

$T(x,y,z)=100e^{\text{−}x-y};$ S consists of the faces of cube $\left|x\right|\le 1,\left|y\right|\le 1,\left|z\right|\le 1.$

$T(x,y,z)=\text{−}\text{ln}\left(x^2+y^2+z^2\right);$ S is sphere $x^2+y^2+z^2=a^2.$

Find the total flux across S with $p=0.$

Show that for $p=3$ the flux across S is independent of a and b .