To determine the center of mass of an object which extends from x = 0 to x = 3, with density function `delta(x), we first find the mass and torque about x = 0 of a slice of length `dxi at position x = xi.
The details, up to the Riemann sums, are as indicated below.
The Riemann sums approach the indicated integrals, which are easily evaluated to obtain total mass 12 and total torque 24.75.
We calculate the center of mass of the triangular region indicated in the figure below.
To determine the width wi, we use similar triangles as indicated.
Recalling that the mass density is 1, the mass of the strip will simply be the area wi * `dxi of the strip, which will give us the expression in the figure below.
The torque will simply be equal to the product of the 'leverage factor' xi and the mass mi, giving is the torque expression indicated below.
The resulting Riemann sums will approach the integrals indicated in the figure below.
The details of the integration and the calculation of the center of mass are shown below.
The figure below depicts a <strong>shed </strong>whose <strong>uniform cross-section </strong>is<strong>part </strong>of a <strong>circle </strong>of radius r centered at the indicatedpoint, and whose <strong>length </strong>is <strong>L</strong>. <ul> <li>We will consider the shed to be <strong>filled </strong>with a <strong>material </strong>whose <strong>density changes </strong>with <strong>height </strong>from the floor.</li></ul></font><blockquote> <p><img src=" cal28.jpg" alt="cal28.jpg (20455 bytes)" WIDTH="276" HEIGHT="174">
We will determine the area of a horizontal strip of thickness `dyi at vertical position yi, as measured from the center of the circle.
By the geometry of the circle, using the Pythagorean Theorem we easily find that half the width is `sqrt(r^2 - (yi + r/4) ^ 2).
The density function in this case has density proportional to the distance from the peak of the shed roof.
We obtain the mass integral indicated in the figure below.
We finally determine the work required to raise a chain, which is initially hanging by one end from the edge of a tall table, to the surface of the table.
Every time we raise a bit of the chain, it gets easier to raise the next bit, so the work done per unit of length decreases as we raise the chain.
To raise the chain at this position, we must raise its weight, which is 2.5 lbs / ft times the length yi of the hanging portion, through a distance `dyi.
