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- 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”.)One-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 ≤ 1One-dimensional objects Find the mass and center of mass of the thin rods with the following density function. ρ(x) = 2 + cos x , for 0 ≤ x ≤ π
- 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 ≤ πAn 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)Differential Calculus: 1. Show or prove that the degree of the HDE is 6. 2.Derive the integrable functions(variable substitution). 3.Find the general solution.
- Bounded 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.Region B: Computing the integral of the function f (x, y) = (x + y) cos (x + y), with a triangle consisting of vertices (0,0), (a, a) and (a, -a).Finding a center of mass Find the center of mass of a thin plateof density d = 3 bounded by the lines x = 0, y = x, and the parabolay = 2 - x2 in the first quadrant.
- Integral Calculus Find the area bounded by the curves y=x^3-18x and x+2y=0Python: Simpson's rule says the integral from x_0 to x_2 of f(x)dx is approximately h(1/3 f(x_0) + 4/3 f(x_1) _1/3 f(x_2)) where h = x_2-x_0 and x_1 is the midpoint of x_0 and x_2. Write a function simp(f,a,b,n) which integrates the function f(x) over the interval [a,b] by dividing it into n subintervals. integrate e^-x over [0,1] to make sure it matches the integral found with scipy.integrate.quad to 5 decimal placesWaiting for the Train. A commuter train arrives punctually at a station every half hour. Each morning, a commuter named John leaves his house and casually strolls to the train station. The time, in minutes, that John waits for the train is a variable with density curve y = 1/30 for 0< x< 30, 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 0 and 30 equals x/30. What percentage of the time does John wait for the trainc. less than 5 minutes?d. between 10 and 15 minutes?e. at least 20 minutes?