# 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 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.

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#### 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.