Part 1. Setup the integral that will give the mass of a metal rod that is 1l inches long (orientated on the r-axis starting at z = 1) if the density of the rod is given by p(z) = Iblin.
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- One-dimensional objects Find the mass and center of mass of the thin rods with the following density function. ρ(x) = 1 + sin x, for 0 ≤ x ≤ πOne-dimensional objects Find the mass and center of mass of the thin rods with the following density function. ρ(x) = 1 + x3, for 0 ≤ x ≤ 1Bounded by the cycloid and the x-axis the density function of the mass of the planar plate placed in the region is constant 1 let it be. Using the second Pappus-Guldin Theorem, we can determine the center of gravity of this plate. Find it.
- An article describes a model for the movement of a particle. Assume that a particle moves within the region A bounded by the x-axis, the line x = 1, and the line y = x. Let(X, Y ) denote the position of the particle at a given time. The joint density of X and Y is given by the function below. Find P(0.5 < X < 1, 0 < Y < 0.5)Hydrodynamic maths obeying Boyle's law, is in motion in a uniform tube of small section, prove that if ? (rho) be the density and v the velocity at a distance x from a fixed point at time t,Dispensing Coffee. A coffee machine is supposed to dispense 6 fluid ounces (fl oz) of coffee into a paper cup. In reality, the amounts dispensed vary from cup to cup. In fact, the amount dispensed, in fl oz, is a variable with density curve y = 2 for 5.75<x< 6.25, and y = 0 otherwise. a. Graph the density curve of this variable.b. Show that the area under this density curve to the left of any number x between 5.75 and 6.25 equals 2x - 11.5. What percentage of cups dispensed by this machine contain c. less than 6 fl oz?d. between 5.9 and 6.1 fl oz?e. at least 5.8 fl oz?
- A shape of a university campus is a square with side length of 10 miles. If you imagine the university on the x-y plane, in the first quadrant with two sides of the square on the positive axes, then the student union is at the origin (at a corner of the square). At noon on a certain day an announcement was made that all students had to walk to the student union. At that time the density function of the students spread over campus was given by: f(x, y) = (3/20000)*(x^2 + y^2 ) 0 < x < 10, 0 < y < 10, 0 otherwise. If students are only allowed to walk parallel to the axes what is the expected value of the distance walked to the student union by a randomly chosen student on campus? (Assume that students walk in a way that monotonically decreases their distance to the student union. That is, they don’t walk ” backward”.)Consider the center of mass of the following. A lamina occupies the part of the disk x2 + y2 ≤ 36 in the first quadrant. The density at any point is proportional to its distance from the x-axis. Find the density function. (Use k as the constant of proportionality.) Find the mass of the entire lamina. (Use k as the constant of proportionality.) Find the moment of the entire lamina with respect to each axis. (Use k as the constant of proportionality.) Find the center of massDetermine the moment of inertia Izz of the torus. The mass of the torus is m and the density ρis constant. Use integral calculus to find your answer. Use shell element. the diameter is 'a'.
- The density Dof an object with mass M and volume Vis D = M/V Determine the density of an object with a mass of 200g and volume of 10 cm ^ 3Astronomers use a technique called stellar stereography todetermine the density of stars in a star cluster from theobserved (two-dimensional) density that can be analyzedfrom a photograph. Suppose that in a spherical cluster ofradius R the density of stars depends only on the distancefrom the center of the cluster. If the perceived star density isgiven by y(s) , where is the observed planar distance fromthe center of the cluster, and x (r ) is the actual density, it canbe shown that y(s) = integral s to r 2 r /sqrt ( r2 - s2 ) x(r) dr If the actual density of stars in a cluster is x (r ) = 1/2 (R-r)2 ,find the perceived density y(s)Two-dimensional plates Find the mass and center of mass of the thin constant-density plates shown in the figure.