Multiple Interpretations of Some Integrals

Abstract

INTRODUCTION In teaching the calculus sequence one talks about centroids, moments of inertia and radii of gyration; volumes of revolution; pressure and total force on submerged planar areas; moments of force and the center of fluid pressure and perhaps even the products of inertia. These integrals are related to one another. Collected here together are several isolated results which do not appear, to my knowledge, in any one book and a new result about the center of fluid pressure coordinate other than the depth of the center of fluid pressure, which is the one usually discussed. It is assumed that the necessary definitions of centroids; moments of area, volume, force and inertia; radius of gyration; and the center of fluid pressure are the usual ones.(3 ) Suppose A is the area of one side of the enclosed region, R, lying wholly within the first quadrant, which is completely submerged in a fluid of uniform volume density, W. If the y-axis lies in the surface of the liquid and the positive x-axis is in the direction of gravity, then the fluid pressure per unit area at a depth x below the surface is given by Wx.(:) The following integrals are meaningful. (See Figure 1.)