Question

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.

Answer #1

1- Set up the triple integral for the volume of the sphere Q=8
in rectangular coordinates.
2- Find the volume of the indicated region.
the solid cut from the first octant by the surface z= 64 - x^2
-y
3- Write an iterated triple integral in the order dz dy dx for
the volume of the region in the first octant enclosed by the
cylinder x^2+y^2=16 and the plane z=10

1a. Using rectangular coordinates, set up iterated integral that
shows the volume of the solid bounded by surfaces z= x^2+y^2+3,
z=0, and x^2+y^2=1
1b. Evaluate iterated integral in 1a by converting to polar
coordinates
1c. Use Lagrange multipliers to minimize f(x,y) = 3x+ y+ 10 with
constraint (x^2)y = 6

Find the surface area of the portion of the plane 3x+2y+z=6 that
lies in the first octant

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

Set-up, but do not evaluate, an iterated integral in polar
coordinates for ∬ 2x + y dA where R is the region in the xy-plane
bounded by y = −x, y = (1 /√ 3) x and x^2 + y^2 = 3x. Include a
labeled, shaded, sketch of R in your work.

Set up an iterated integral for the triple integral in spherical
coordinates that gives the volume of the hemisphere with center at
the origin and radius 5 lying above the xy-plane.

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

Evaluate the surface integral.
5. " S x 2 z dσ; S that part of the cylinder x 2 + z 2 = 1 which
lies between the planes y = 0 and y = 2, and is above the
xy-plane.

Set up iterated integrals for both orders of integration. Then
evaluate the double integral using the easier order.
y dA, D is bounded by y = x
− 20; x = y2
D

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

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