Relates Line Integral to the Polar Moment of Inertia (PMOI)

Consider the vector field F=((x^2)*y+(y^3)/3)i,(i is the horizontal unit vector) and let C be the portion of the graph y=f(x) running from (x1,f(x1)) to (x2,f(x2)) (assume that x1<x2, and f takes positive values).

Show that the lineintegral "integral(F.dr)" is equal to the polar moment of inertia of the region R lying below C and above the x-axis.

Solution Preview

Consider the vector field F=((x^2)*y+(y^3)/3)i,(i is the horizontal unit vector) and let C be the portion of the graph y=f(x) running from (x1,f(x1)) to (x2,f(x2)) (assume that x1<x2, and f takes positive values).

Show that the line integral "integral(F.dr)" is equal to the polar moment of inertia of the region R lying below C and above the x-axis. ...

Solution Summary

This solution explains, step-by-step, how a Line Integral of a Vector Field along a given Contour C can be shown to be related to the Double Integral over a Region R using the Green's Theorem and hence can be proved to be equal to the Polar Moment of Inertia of the region R lying below C and above the x-axis.

Using the coordinate change u=xy, v=y/x, set up an iterated integral for thepolarmoment of inertia of the region bounded by the hyperbola xy=1 , the x-axis, and the two lines x=1 and x=2.
Choose the order of integration which make the limits simplest
THIS MESSAGE IS ADDRESSED TO ANY TA:
I found something , I just want you

See the attached file.
1. Evaluate theintegral:
** see the attachment for the full equation **
Where D is the domain given by: 1 ≤ x^2 + y^2 ≤ 4 and y ≥ 0.
2. (i) Find the area of the region enclosed by the ellipse (x^2/a^2) + (y^2/b^2) = 1
(ii) Find the area of the region enclosed by the parabola y = x^2 an

A sphere consists of a solid wooden ball of uniform density 800kg/m^3 and radius 0.20 m and is covered with a thin coating of lead foil with area density 20kg/m^2 .
Calculate themoment of inertia of this sphere about an axis passing through its center.

Prim is primitive!
In genral themoment of inertia around an axis( a line) L is:
Isubl=double prim (dist(.,L)^2*delta*dA)
The collection of lines parallel to the y axis have the form x=a .Let I=Isub(y) be the usual moment of inertia around the y axis
I= double prim of x^2*delta*dA
Let I(bar) be themoment of ine

(See attached file for full problem description with equations)
(Steiner's theorem) If IA is themoment of inertia of a mass distribution of total mass M with respect to an axis A through the center of gravity, show that its moment of inertia IB with respect to an axis B, which is parallel to A and has the distance k from it,

A figure skater during her finale can increase her rotation rate from an initial rate of 1.12 revolutions every 1.88 s to a final rate of 3.15 revolutions per second. If her initial moment of inertia was 4.73 kg* m2, what is her final moment of inertia?

A sheet of plywood 1.37 cm thick is used to make a cabinet door 50.9 cm wide by 76.9 cm tall, with hinges mounted on the vertical edge. A small 191-g handle is mounted 45 cm from the lower hinge at the same height as that hinge. If the density of the plywood is 550 kg/m^3, what is themoment of inertia of the door about the hing

A cylindrical fishing reel has a moment of inertia of I = 6.81à?10^-4 kg* m2 and a radius of 3.13 cm. A friction clutch in the reel exerts a restraining torque of 1.49 N*m if a fish pulls on theline. The fisherman gets a bite, and the reel begins to spin with an angular acceleration of 63.7 rad/s^2.
1) What is the force of