Question

Consider the part of the paraboloid x = 4 − y ^2 − z ^2 that lies in front of the plane x = 0.

(a) What is its mass if its density is ρ(x, y, z) = y^2 + z ^2 g/cm^2 ?

(b) What is its surface area?

Answer #1

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

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...

Given S is the surface of the paraboloid z= 4-x^2-y^2
and C is the curve of intersection of the paraboloid with the plane
z=0. Verify stokes theorem for the field F=2zi+xj+y^2k.( you might
verify it by checking both sides of the theorem)

Find the area of the surface
The part of the parabloid z=4-x^2-y^2 that lies above the
xy-plane

Consider the surface given by the part of the plane z=y+3 that
lies inside the cylinder x^2+y^2=4
a. Find a parametric representation of the surface
b. find the area of the above surface

7. What is the surface area of the paraboloid parametrized in
problem 5(b)?
(The paraboloid from problem 5(b) is z = 4 - x^2 - y^2 which
lies above the xy-plane)

Use Stokes" Theorem to evaluate (F-dr where F(x, y, z)=(-y , x-z
, 0) and the surface S is the part of the paraboloid : z = 4- x^2 -
y^2 that lies above the xy-plane. Assume C is oriented
counterclockwise when viewed from above.

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.

Find the volume of the solid that lies under the paraboloid z =
x^2 + y^2 , above the xy-plane and inside the cylinder x^2 + y^2 =
1.

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=

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