Consider the unit sphere x2 +y2 +z2 = 1 and the cone (z+√2)2 =
x2 +y2....
Consider the unit sphere x2 +y2 +z2 = 1 and the cone (z+√2)2 =
x2 +y2. Show that these surfaces are tangent where they intersect,
that is, for a point on the intersection, these surfaces have the
same tangent plane
The plane x+y+z= 24 intersects the cone
x2+y2= z2 in an ellipse. The goal
of this exercise is to find the two points on this ellipse that are
closest to and furthest away from the xy-plane. Thus, we want to
optimize F(x,y,z)= z, subject to the two constraints G(x,y,z)=
x+y+z= 24 and H(x,y,z)= x2+y2-z2=
0.
Let H be the hemisphere x2 + y2 + z2 = 66, z ≥ 0, and suppose f
is a continuous function with f(4, 5, 5) = 5, f(4, −5, 5) = 11,
f(−4, 5, 5) = 12, and f(−4, −5, 5) = 15. By dividing H into four
patches, estimate the value below. (Round your answer to the
nearest whole number.) H f(x, y, z) dS
Let F= (x2 +
y + 2 + z2) i + (exp(
x2 ) + y2)
j + (3 + x) k . Let a
> 0 and let S be part of the spherical
surface x2 + y2 +
z2 = 2az + 15a2
that is above the x-y plane and the disk formed in the
x-y plane by the circular intersection between the sphere
and the plane. Find the flux of F outward across
S.
Evaluate the following integral,
∫
∫
S
(x2 + y2 + z2) dS,
where S is the part of the cylinder x2 +
y2 = 64 between the planes z = 0 and
z = 7, together with its top and bottom disks.
If C is the part of the circle (x2)2+(y2)2=1(x2)2+(y2)2=1 in the
first quadrant, find the following line integral with respect to
arc length.
∫C(8x−6y)ds