CalculusVolume3-OP_mktoy8b-159

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6.60 A cast-iron solid ball is given by inequality x2 + y2 + z2 ≤ 1. The temperature at a point in a region
containing the ball is T(x, y, z) = 1
3
⎛
⎝x2 + y2 + z2⎞
⎠. Find the heat flow across the boundary of the solid if this
boundary is oriented outward.
Chapter 6 | Vector Calculus 783
6.6 EXERCISES
For the following exercises, determine whether the
statements are true or false.
269. If surface S is given by
⎧
⎩
⎨(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, z = 10⎫
⎭
⎬, then
∬
S
f (x, y, z)dS = ∫
0
1
∫
0
1
f ⎛
⎝x, y, 10⎞
⎠dxdy.
270. If surface S is given by
⎧
⎩
⎨(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, z = x⎫
⎭
⎬, then
∬
S
f (x, y, z)dS = ∫
0
1
∫
0
1
f (x, y, x)dxdy.
271. Surface
r = 〈 v cos u, v sin u, v2 〉 , for 0 ≤ u ≤ π, 0 ≤ v ≤ 2,
is the same as surface r = 〈 v cos 2u, v sin 2u, v 〉 ,
for 0 ≤ u ≤ π
2, 0 ≤ v ≤ 4.
272. Given the standard parameterization of a sphere,
normal vectors tu × tv are outward normal vectors.
For the following exercises, find parametric descriptions
for the following surfaces.
273. Plane 3x − 2y + z = 2
274. Paraboloid z = x2 + y2, for 0 ≤ z ≤ 9.
275. Plane 2x − 4y + 3z = 16
276. The frustum of cone z2 = x2 + y2, for 2 ≤ z ≤ 8
277. The portion of cylinder x2 + y2 = 9 in the first
octant, for 0 ≤ z ≤ 3
278. A cone with base radius r and height h, where r and
h are positive constants
For the following exercises, use a computer algebra system
to approximate the area of the following surfaces using a
parametric description of the surface.
279. [T] Half cylinder
{(r, θ, z) : r = 4, 0 ≤ θ ≤ π, 0 ≤ z ≤ 7}
280. [T] Plane z = 10 − x − y above square
|x| ≤ 2, |y| ≤ 2
For the following exercises, let S be the hemisphere
x2 + y2 + z2 = 4, with z ≥ 0, and evaluate each
surface integral, in the counterclockwise direction.
281. ∬
S
zdS
282. ∬
S
(x − 2y)dS
283. ∬
S
⎛
⎝x2 + y2⎞
⎠zdS
For the following exercises, evaluate ∫ ∫
S
F · Nds for
vector field F, where N is an outward normal vector to
surface S.
784 Chapter 6 | Vector Calculus
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284. F(x, y, z) = xi + 2yj − 3zk, and S is that part of
plane 15x − 12y + 3z = 6 that lies above unit square
0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
285. F(x, y, z) = xi + yj, and S is hemisphere
z = 1 − x2 − y2.
286. F(x, y, z) = x2 i + y2 j + z2 k, and S is the portion
of plane z = y + 1 that lies inside cylinder x2 + y2 = 1.
For the following exercises, approximate the mass of the
homogeneous lamina that has the shape of given surface S.
Round to four decimal places.
287. [T] S is surface
z = 4 − x − 2y, with z ≥ 0, x ≥ 0, y ≥ 0; ξ = x.
288. [T] S is surface z = x2 + y2, with z ≤ 1; ξ = z.
289. [T] S is surface
x2 + y2 + x2 = 5, with z ≥ 1; ξ = θ2.
290. Evaluate ∬
S
⎛
⎝y2 zi + y3 j + xzk⎞
⎠ · dS, where S is the
surface of cube
−1 ≤ x ≤ 1, −1 ≤ y ≤ 1, and 0 ≤ z ≤ 2. in a
counterclockwise direction.
291. Evaluate surface integral ∬
S
gdS, where
g(x, y, z) = xz + 2x2 − 3xy and S is the portion of plane
2x − 3y + z = 6 that lies over unit square R:
0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
292. Evaluate ∬
S
(x + y + z)dS, where S is the surface
defined parametrically by
R(u, v) = (2u + v)i + (u − 2v)j + (u + 3v)k for
0 ≤ u ≤ 1, and 0 ≤ v ≤ 2.
293. [T] Evaluate ∬
S
(x − y2 + z)dS, where S is the
surface defined by
R(u, v) = u2 i + vj + uk, 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.
294. [T] Evaluate where S is the surface defined by
R(u, v) = ui − u2 j + vk, 0 ≤ u ≤ 2, 0 ≤ v ≤ 1. for
0 ≤ u ≤ 1, 0 ≤ v ≤ 2.
295. Evaluate ∬
S
⎛
⎝x2 + y2⎞
⎠dS, where S is the surface
bounded above hemisphere z = 1 − x2 − y2, and below
by plane z = 0.
Chapter 6 | Vector Calculus 785
296. Evaluate ∬
S
⎛
⎝x2 + y2 + z2⎞
⎠dS, where S is the
portion of plane z = x + 1 that lies inside cylinder
x2 + y2 = 1.
297. [T] Evaluate ∬
S
x2 zdS, where S is the portion of
cone z2 = x2 + y2 that lies between planes z = 1 and
z = 4.
298. [T] Evaluate ∬
S
⎛
⎝xz/y⎞
⎠dS, where S is the portion of
cylinder x = y2 that lies in the first octant between planes
z = 0, z = 5, y = 1, and y = 4.
299. [T] Evaluate ∬
S
(z + y)dS, where S is the part of
the graph of z = 1 − x2 in the first octant between the
xz-plane and plane y = 3.
300. Evaluate ∬
S
xyzdS 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).
301. Find the mass of a lamina of density ξ(x, y, z) = z
in the shape of hemisphere z = ⎛
⎝a2 − x2 − y2⎞
⎠
1/2
.
302. Compute ∫ ∫
S
F · NdS, where
F(x, y, z) = xi − 5yj + 4zk and N is an outward normal
vector S, where S is the union of two squares
S1 : x = 0, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 and
S2 : z = 1, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
786 Chapter 6 | Vector Calculus
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303. Compute ∫ ∫
S
F · NdS, where
F(x, y, z) = xyi + zj + (x + y)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.
304. Compute ∫ ∫
S
F · NdS, where
F(x, y, z) = 2yzi + ⎛
⎝tan−1 xz⎞
⎠j + exy k and N is an
outward normal vector S, where S is the surface of sphere
x2 + y2 + z2 = 1.
305. Compute ∫ ∫
S
F · NdS, where
F(x, y, z) = xyzi + xyzj + xyzk and N is an outward
normal vector S, where S is the surface of the five faces
of the unit cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 missing
z = 0.
For the following exercises, express the surface integral as
an iterated double integral by using a projection on S on the
yz-plane.
306. ∬
S
xy2 z3 dS; S is the first-octant portion of plane
2x + 3y + 4z = 12.
307. ∬
S
⎛
⎝x2 − 2y + z⎞
⎠dS; S is the portion of the graph of
4x + y = 8 bounded by the coordinate planes and plane
z = 6.
For the following exercises, express the surface integral as
an iterated double integral by using a projection on S on the
xz-plane
308. ∬
S
xy2 z3 dS; S is the first-octant portion of plane
2x + 3y + 4z = 12.
309. ∬
S
⎛
⎝x2 − 2y + z⎞
⎠dS; S is the portion of the graph of
4x + y = 8 bounded by the coordinate planes and plane
z = 6.
310. Evaluate surface integral ∬
S
yzdS, where S is the
first-octant part of plane x + y + z = λ, where λ is a
positive constant.
311. Evaluate surface integral ∬
S
⎛
⎝x2 z + y2 z⎞
⎠dS,
where S is hemisphere x2 + y2 + z2 = a2, z ≥ 0.
312. Evaluate surface integral ∬
S
zdA, where S is
surface z = x2 + y2, 0 ≤ z ≤ 2.
313. Evaluate surface integral ∬
S
x2 yzdS, where S is
the part of plane z = 1 + 2x + 3y that lies above rectangle
0 ≤ x ≤ 3 and 0 ≤ y ≤ 2.
314. Evaluate surface integral ∬
S
yzdS, where S is plane
x + y + z = 1 that lies in the first octant.
315. Evaluate surface integral ∬
S
yzdS, where S is the
part of plane z = y + 3 that lies inside cylinder
x2 + y2 = 1.
For the following exercises, use geometric reasoning to
evaluate the given surface integrals.
316. ∬
S
x2 + y2 + z2dS, where S is surface
x2 + y2 + z2 = 4, z ≥ 0
317. ∬
S
(xi + yj) · dS, where S is surface
x2 + y2 = 4, 1 ≤ z ≤ 3, oriented with unit normal
vectors pointing outward
318. ∬
S
(zk) · dS, where S is disc x2 + y2 ≤ 9 on
plane z = 4, oriented with unit normal vectors pointing
upward
Chapter 6 | Vector Calculus 787