Question

a) Set up the integral in cylindrical coordinates for the moment
of inertia about the z-axis for a cone bounded by z = sqrt x^2+y^2
and z = 2. The density of the cone is a constant 5.

b) Evaluate the integral by using direct method the work done by ~
F(x,y) = (y,−x) along ~r(t) = (2cos(t),2sin(t)), 0 ≤ t ≤ π.

need help on a and b

Answer #1

sir work done coming negative due to may be work done to system. If any mistake plz comment

Set up a triple integral in cylindrical coordinates to compute
the volume of the solid bounded between the cone z 2 = x 2 + y 2
and the two planes z = 1 and z = 2.
Note: Please write clearly. That had been a big problem for me
lately. no cursive Thanks.

Set up (Do Not Evaluate) a triple integral that yields the
volume of the solid that is below
the sphere x^2+y^2+z^2=8
and above the cone z^2=1/3(x^2+y^2)
a) Rectangular coordinates
b) Cylindrical
coordinates
c) Spherical
coordinates

Write down a cylindrical coordinates integral that gives the
volume of the solid bounded above by z = 50 − x^2 − y^2 and below
by z = x^2 + y^2 . Evaluate the integral. (Hint: use the order of
integration dz dr dθ.)

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.

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.

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 moment of inertia about x-axis of lamina with shape of the
region bounded by y=x^2 y=0 and x=0 with density=x+y

Use a detailed analysis to set up but not evaluate an integral
for the volume Z of the solid generated by revolving the region
bounded by the curves 2x = y^2, x = 0, and y = 4 about the
y-axis.

7. Given The triple integral E (x^2 + y^2 + z^2 ) dV where E is
bounded above by the sphere x 2 + y 2 + z 2 = 9 and below by the
cone z = √ x 2 + y 2 . i) Set up using spherical coordinates. ii)
Evaluate the integral

Set up, but do not evaluate, the integral for the volume of the
solid obtained by rotating the region enclosed by y=\sqrt{x}, y=0,
x+y=2 about the x-axis. Sketch
a) By Washers
b) Cylindrical shells

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