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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

### Solutions to Various First Order ODEs

Solve using integrating factor and Bernoullis.

### 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

### Solutions to Several First Order ODEs

Solve 1st order DE, please include explaination of process, if possible. Thanks

### 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

### Multiple integrals are examined.

Express D as a union of regions of type I or type II and evaluate the integral. Problem is attached.

### 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