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Integrals

finite integrals domain

Prove that a ?nite integral domain is a ?eld. Give an example to show that an in?nite integral domain need not be a ?eld. (Hint: Given a â??R consider the map R â?'R de?ned by x â?'ax. Is it injective? Surjective?)

Compute the expectation of random variables.

Exercise 1: Let Y be a random variable with normal distribution with mean (μ =2), and standard deviation (σ = 1). Let J(t) = Y I[1,∞)(t). (I =1 in the interval [1,∞), and 0 otherwise). Define a new random variable L by putting that L is equal to integral from minus infinity to plus infinity e^t dJ(t). Compute the expectat

Chain rule example problem

(a) Show that the change of variable v = ln(y) transforms the differential equation (dy/dx) + P(x)((y)ln(y)) = Q(x)y into the linear equation (dv/dx) + P(x)v = Q(x) (b) Use the idea of part (a) to solve the differential equation x(dy/dx) + 2y(ln(y)) = 4(x^3)y

Calculating integrals

Q1a)Find The area of the paraboloid x^2+y^2=z inside the cylinder x^2+y^2=9 b)write a triple integral in cylindrical coordinates for the volume inside the cone z^2=x^2+y^2 and between the planes z=1 ans z=2 c)Find the moment of inertia of a circular disk (uniform density) about an axis through its center and perpen

Find the average value of spring displacement

** Please see the attachment for the complete problem description ** A retarding force, symbolized by the dashdot in the figure to the right, slows the motion of the weighted spring so that the mass's position at time t is y = 22e^(-t) cos t, t >=0. Find the average value of y over the interval 0 < t < pi

Fluid Volume Calculation

Please see the attachment. The end plates (isosceles triangles) of the trough shown below were designed to withstand a fluid force of 6000 lb. How many cubic feet can the tank hold without exceeding this limitation? Assuming the density is 62.4 lb/ft^3, the maximum volume is ? ft^3.

Methods of approximating integrals are depicted.

What are some methods to approximate the value of an integral when it cannot be calculated directly? Show how each method works on a problem that can be solved directly, and compare the results, including the error estimations of the approximation methods.

Integration Using the Substitution Method

** Please see the attached file for the complete problem description ** Please complete #6 6 in this attached problem. Use the substitution formula to evaluate: integral sign 6x/sqrt(x^2 + 4) dx

Step-by-step Integration of Functions

Integrate the following functions. Show the work, including method used in detail. 1. f(x)=2(cos(x+1))^2 2. F(x) = sin[2x]*e^(-x) +1 ** Please see the attached file for a Word formatted copy of the problem.

Integral of Upper and Lower Limits

Evaluate the definite integral and round the solution to three decimal places. Evaluate the integral and round to three decimal places. ∫ (upper limit on the integral symbol is 2 and lower is 1) [4√x-5/x]dx.

The velocity of a particle

(a) Consider the three vectors a = i - 2j + k, b = 2i + 4k and c = i + 2j + 3k. i. Find the sum a+b, the scalar product a.b and the vector product axb. ii. Are the three vectors a, b and c co-planar? Explain your reasoning. (b) The velocity of a particle at time t is v(t) = sin ti + cos tj - 9.8 tk. i. What is

Integration Question for Probability/Calculus

For every one-dimensional set C for which the integral exists, let Q(C) = &#8747;c f(x) dx , where f(x) = 6x(1 - x) , 0 < x < 1, zero elsewhere, otherwise let Q(C) be undefined. If C1 = { x : ¼ < x < ¾ } , C2 = {1/2}, C3 = {x: 0 < x < 10}. Find Q(C1), Q(C2) and Q(C3). Without doing any work I would

Variation of parameter formula

See the attached problem. Use the formula stated only to solve. Show steps clearly. USING ONLY THE VARIATION OF PARAMETER FORMULA: Find the particular integral for each of the equations below: 1) 2) Show each step clearly

Solve the IBVP for the heat equation.

Please show all steps. Thank you. Solve the IBVP for the heat equation u_t = u_xx, 0 < x < pi, with Neumann boundary conditions u_x(0, t) = 1, u_x(pi, t) = 0, and initial condition u(x, 0) = 0. Hint. reduce to homogenous boundary conditions by subtracting a function U(x) that satisfies U_x(0) = 1, U_x(pi) = 0.

Integral domains, Principle Ideal domains, Fields

For each of the following rings answer the following questions: 1. Is it an integral domain? 2. Is it a principal ideal domain (PID)? 3. Is it a field? Give reasons (i.e. short proofs, if needed) for your answer. 1. Z/13Z; 2. Z/20Z; 3. ZÃ?Z with componentwise addition and multiplication; 4. Q[X]/(f) with f=X^2+X+

Ordinary differential equation.

Can you explain how you get from: dx derivative of (x^2 +9), divided by square root of (x^2+9) to: 3 sec^2 divided by 3 sec? I was given it as part of an answer to the ode: square root of x^2+9 dy/dx = y^2. Thank you.

Kronecker formula for integration

** Please see the attached file for a Word formatted copy of the problem ** Consider the function f(x) = cos x , 0 < x < π, as a periodic function of period π. Plot the function on -2π < x< 2π, and find its Fourier series. Now consider the odd periodic extension of f(x). Plot f(x) on -2π < x< 2π and find its Four

definition of Lebesgue sum

Using the definition of Lebesgue sum, show Lebesgue integral. 21.1 Definition: If f is bounded measurable function on a bounded measurable set , if is a partition of and if for then we call a Lebesgue sum of f relative to P. 26.5 Show directly from Definition 21.1 that if f is bounded and m(A)=0, then . 26.7 S

circumference and area enclosed

Find the circumference and area enclosed by the casing of a Wankel engine, which is a curve with the parametric equations : { x = 2cos(3t) + 6cos(t) { y = 2 sin(3t) + 6sin(t) where 0 â?¤ t â?¤ 2Pi

Inner product

Consider the functions below and the corresponding inner product are these functions orthogonal to each other with respect to the inner product? f(x)=cos x g(x)=(3 cos ^2 x-1) {INTEGRAL sign from 0 to pi} f(x) g(x) sin x dx

approximations to different integrals

Below are some approximations to different integrals. By extrapolating from these approximations give the value of the integral to a level of accuracy you feel is justified. • T2=4.02441, T4=4.09123. • M8=11.3421, M16=10.6745. • S1=4.4692, S2=4.9899.

Prove That Ind (Gamma) Is Always an integer

Let gamma be a continuously differentiable closed curve in the complex plane, with parameter interval [a,b], and assume that gamma (t) is not equal to 0 for every t in [a,b]. Define the index of gamma to be Ind (gamma) = 1/2pii integral from a to b of gamma prime (t) over gamma (t) dt. Prove that Ind (gamma) is always an integer

The Body Travels Only an Infinite Distance

Suppose that a body moves in a straight line through a resisting medium with resistance proportional to the square of it's speed v with initial speed v_0 and initial position x_0. Show that it's speed and position are given by v(t)=v_0/(1+v_0kt), x(t)=x_0+((ln(1+v_0kt))/k). Conclude that the body travels only an infinite distanc

Integration problems

I need help solving the following problems. Can you explain to me how to solve these problems? Find the derivative of y = log(2 + sin⁡〖x)〗 Find the derivative of f(x) = sinh2(3x) Find the derivative of y = x^(x-1) Find the derivative of y = ln⁡((1+e^x)/(1-e^x )) Evaluate ∫_0^(π/4)▒tan⁡

The Volume of the Solid by Rotating the Region around Y-Axis

Please explain in detail as much possible: Find the volume of a triangle with sides y = 2, y = x+1, y = 11-x. The vertices are (5, 6), (1, 2) and (9, 2). Find the volume by rotating about y-axis: (i) Method of disks (ii) Cylindrical shell method

a first order non homogenous equation

A tank contains 200 liters of fluid n which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 4L/min; the well-mixed solution is pumped up at the same rate. Find the number A(t) of grams of salt in the tank at time t.

Equilibrium solutions, Separation of variables, exact equations,

What technique would be used to solve the differential equation: You do not have to solve the differential equations, just write what technique that you would use to solve them. Solution techniques: Equilibrium solutions, Separation of variables, exact equations, integrating factors, substitutions for homogeous and Bernoulli