#ISOSCELES TRAPEZOID CALCULATOR EXAMPLES LICENSE#
Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License The partition divides the plate into several thin, rectangular strips (see the following figure). For i = 0, 1, 2 ,…, n, i = 0, 1, 2 ,…, n, let P = be a regular partition of the interval, , and for i = 1, 2 ,…, n, i = 1, 2 ,…, n, choose an arbitrary point x i * ∈. Because density is a function of x, x, we partition the interval from along the x -axis. Then, the density of the disk can be treated as a function of x, x, denoted ρ ( x ). We orient the disk in the x y -plane, x y -plane, with the center at the origin. We assume the density is given in terms of mass per unit area (called area density), and further assume the density varies only along the disk’s radius (called radial density). As with the rod we looked at in the one-dimensional case, here we assume the disk is thin enough that, for mathematical purposes, we can treat it as a two-dimensional object. We now extend this concept to find the mass of a two-dimensional disk of radius r. If the density of the rod is given by ρ ( x ) = 2 x 2 + 3, ρ ( x ) = 2 x 2 + 3, what is the mass of the rod? Note that although we depict the rod with some thickness in the figures, for mathematical purposes we assume the rod is thin enough to be treated as a one-dimensional object.Ĭonsider a thin rod oriented on the x-axis over the interval. Orient the rod so it aligns with the x -axis, x -axis, with the left end of the rod at x = a x = a and the right end of the rod at x = b x = b ( Figure 6.48). We can use integration to develop a formula for calculating mass based on a density function. We then turn our attention to work, and close the section with a study of hydrostatic force. Let’s begin with a look at calculating mass from a density function.
In this section, we examine some physical applications of integration. 6.5.5 Find the hydrostatic force against a submerged vertical plate.6.5.4 Calculate the work done in pumping a liquid from one height to another.6.5.3 Calculate the work done by a variable force acting along a line.6.5.2 Determine the mass of a two-dimensional circular object from its radial density function.6.5.1 Determine the mass of a one-dimensional object from its linear density function.
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