Question

Let S be the boundary of the solid bounded by the paraboloid
z=x^2+y^2 and the plane z=16

S is the union of two surfaces. Let S1 be a portion of the plane
and S2 be a portion of the paraboloid so that S=S1∪S2

Evaluate the surface integral over S1

∬S1 z(x^2+y^2) dS=

Evaluate the surface integral over S2

∬S2 z(x^2+y^2) dS=

Therefore the surface integral over S is

∬S z(x^2+y^2) dS=

Answer #1

In the following problems, the surface S is the part of the
paraboloid z= x^2 + y^2 which lies below the plane z= 4, and
includes the circular intersection with this plane. This single
surface S could also be described as being contained inside the
cylinder x^2+y^2= 4.
(a) Iterate, but do not evaluate, the integral ∫∫S(z+x) dS in
terms of two parameters. Write the integrand in simplest form.
(b) Use Stoke’s theorem to rewrite ∫S(delta X F) · ndS...

Lets consider the solid bounded above a sphere x^2+y^2+z^2=2 and
below by the paraboloid z=x^2+y^2.
Express the volume of the solid as a triple integral in
cylindrical coordinates. (Please show all work clearly) Then
evaluate the triple integral.

4. Consider the solid bounded by the paraboloid x^2+ y^2 + z = 9
as well as by the planes y = 3x and z = 0 in the first octant.
(a) Graph the integration domain D.
(b) Calculate the volume of the solid with a double
integral.

Let S1: x^2+y^2=4 and S2: z=−√(x^2+y^2) be two surfaces in
space.
(a) [2] Graph these two surfaces.
(b) [4] Find equations of S1 and S2 in spherical coordinate
system .
(c) [4] Find the intersection of S1 and S2 in this (spherical)
coordinate system.
(d) [5] SET UP but DO NOT EVALUATE the triple integral in
spherical coordinate system to evaluate the volume which is above
the xy -plane, outside of S1 and inside of S2 .
(Bonus) [2] Can...

Problem 10. Let F = <y, z − x, 0> and let S be the surface
z = 4 − x^2 − y^2 for z ≥ 0, oriented by outward-pointing normal
vectors.
a. Calculate curl(F).
b. Calculate Z Z S curl(F) · dS directly, i.e., evaluate it as a
surface integral.
c. Calculate Z Z S curl(F) · dS using Stokes’ Theorem, i.e.,
evaluate instead the line integral I ∂S F · ds.

Q8. Let G be the cylindrical solid bounded by x2 + y2 = 9, the
xy-plane, and the plane
∫∫
z = 2, and let S be its surface. Use the Divergence Theorem to
evaluate I = S F · ndS where F(x,y,z) = x3i + y3j + z3k and n is
the outer outward unit normal to S.

. Find the volume of the solid bounded by the cylinder x 2 + y 2
= 1, the paraboloid z = x 2 + y 2 , and the plane x + z = 5

Use triple integral and find the volume of the solid E bounded
by the paraboloid z = 2x2 + 2y2 and the plane
z = 8.

Find the volume of the region bounded below by the paraboloid z
= x^2 + y^2 and above by the plane z = 2x.

a) Let R be the solid enclosed by the paraboloid z = 8 − (x^2+
y^2) and the cone z=2 sqrt(x^2+y^2) FindthevolumeofR.

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