Question

evaluate area of curved surface paraboloid z=9-x^2-y^2 in the first octant using surface integral

Answer #1

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.

1. Evaluate ???(triple integral) E
x + y dV
where E is the solid in the first octant that lies under the
paraboloid z−1+x2+y2 =0.
2.Evaluate ???(triple integral) square root ?x^2+y^2+z^2 dV
where E lies above the cone z = square root x^2+y^2 and between
the spheres x^2+y^2+z^2=1 and x^2+y^2+z^2=9

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=

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...

Evaluate the surface integral (double integral) over S: G(x,y,z)
d sigma using a parametric description of the surface.
G(x,y,z)= 3z^2, over the hemisphere x^2+y^2+z^2=16, with z
greater than or equal to 0.

Set up an iterated integral for the surface area of the part of
the plane x + y + z = 6 that lies in the first octant.

Find the area of the surface. The part of the paraboloid
z=1-x^2-y^2 that lies above the plane z=-2
(Please post hand writing one) thank you

2. Evaluate the double integral Z Z R e ^(x^ 2+y ^2) dA where R
is the semicircular region bounded by x ≥ 0 and x^2 + y^2 ≤ 4.
3. Find the volume of the region that is bounded above by the
sphere x^2 + y^2 + z^2 = 2 and below by the paraboloid z = x^2 +
y^2 .
4. Evaluate the integral Z Z R (12x^ 2 )(y^3) dA, where R is the
triangle with vertices...

Evaluate the surface integral (x+y+z)dS when S is part of the
half-cylinder x^2 +z^2=1, z≥0, that lies between the planes y=0 and
y=2

Evaluate the surface integral.
S
z +
x2y
dS
S is the part of the cylinder
y2 +
z2 = 4
that lies between the planes
x = 0 and x = 3
in the first octant

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