Question

6. Let R be the tetrahedron in the first octant bounded by the coordinate planes and the plane passing through (1, 0, 0), (0, 1, 0), and (0, 0, 2) with equation 2x + 2y + z = 2, as shown below. Using rectangular coordinates, set up the triple integral to find the volume of R in each of the two following variable orders, but DO NOT EVALUATE.

(a) triple integral 1 dxdydz

(b) triple integral of 1 dzdydx

Answer #1

Let E be the solid that lies in the first octant, inside the
sphere x2 + y2 + z2 = 10. Express the volume of E as a triple
integral in cylindrical coordinates (r, θ, z), and also as a triple
integral in spherical coordinates (ρ, θ, φ). You do not need to
evaluate either integral; just set them up.

Find 6 different iterated triple integrals for the volume of the
tetrahedron cut from the first octant (when x > 0, y > 0, and
z > 0) by the plane 6x + 2y + 3z = 6. Dont evaluate the
integrals.

Use a triple integral to find the volume of the given solid.
The tetrahedron enclosed by the coordinate planes and the
plane
11x + y + z = 2

B is the solid occupying the region of the space in the first
octant and bounded by the paraboloid z = x2 + y2- 1 and the planes
z = 0, z = 1, x = 0 and y = 0. The density of B is proportional to
the distance at the plane of (x, y).
Determine the coordinates of the mass centre of solid B.

Use Divergence theorem to evaluate surface integral S F ·n dA
where S is the surface of the solid enclosed by the tetrahedron
formed by the coordinate planes x = 0, y = 0 and z = 0 and the
plane 2x + 2y + z = 6 and F = 2x i − x^2 j + (z − 2x + 2y) k.

1.Set up the bounds for the following triple integral: R R R E
(2y)dV where E is bounded by the planes x = 0, y = 0, z = 0, and 3
= 4x + y + z. Do NOT integrate.
2.Set up the triple integral above as one of the other two types
of solids E.

(3) Let D denote the disk in the xy-plane bounded by the circle
with equation y2 = x(6−x). Let S be the part of the
paraboloid z = x2 +y2 + 1 that lies above the
disk D.
(a) Set up (do not evaluate) iterated integrals in rectangular
coordinates for the following.
(i) The surface area of S.
(ii) The volume below S and above D.
(b) Write both of the integrals of part (a) as iterated
integrals in cylindrical...

Set up a double integral in rectangular coordinates for the
volume bounded by the cylinders x^2+y^2=1 and y^2+z^x=1 and
evaluate that double integral to find the volume.

let R be the region bounded by the curves x = y^2 and x=2y-y^2.
sketch the region R and express the area R as an iterated integral.
(do not need to evaluate integral)

Consider the plane region R bounded by the curve y = x − x 2 and
the x-axis. Set up, but do not evaluate, an integral to find the
volume of the solid generated by rotating R about the line x =
−1

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