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# Integrals

### Evaluating Triple Integrals

See Attached Question 1: (1 points) Evaluate the following integral Question 2: (1 points) Evaluate the following integral Question 3: (1 points) Evaluate the following integral Question 4: (1 points) Evaluate the following integral Question 5: (1 points) Evalu

### Residue theorem

I need the solution of: integral from 0 to infinity of x^3 sin(2x)/(x^2+1)^2 dx

### The Integration of the Function

given that integral from 0 to infinity of e^(-x^2)dx=sqrt(pi)/2 show by integrating e^(-z^2) around a rectangle with vertices at z=0,p,p+ai and ai (with a positive and letting p go to infinity) that integral from 0 to infinity of e^(-x^2)cos(2ax)dx= [e^(-a^2)]*sqrt(pi)/2 integral from 0 to infinity of e^(-x^2)sin(2ax)dx= [

### Proof that an Integral Is Divergent

f(x) = arctan(x)/ x for x !=0 f(x) = 1 for = 0 f'(x) = 1/(x+x^3) -arctan(x)/x^2 for x !=0 f'(x) = 0 for x = 0 F is the integral from 0 to x of f(x) Know that F(x) is strictly increasing and F(x) = F(-x) Show that lim F(x) = infinity when x goes to infinity

### Applications of Integration

(1) The acceleration due to gravity near the surface of Mars is 3.72ms^-2. A rock is thrown straight up from the surface with an initial velocity of 23ms^-2. How high does it go? (Note: gravitational acceleration acts downwards so is negative). (2) A 30m uniform chain (mass 25kg) is hanging from the roof of a factory. What wo

### Ordinary Differential Equations

Hi there, I have a question which can be located here http://nullspace8.blogspot.com/2011/10/e1_26.html. can someone please take a look? Full working step by step solution in pdf or word please. If you think the bid is insufficient and you can do it, please respond with a counter bid. Thank you.

### 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.

### Defining an Improper Integral

What exactly is an improper integral? Why is there a difference between improper and proper integrals?

### 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

### Step-by-step integration including logarithmic

1. integrate. Log xdx 2. Integrate. X^2x^2-1 3. use log diff to find. Y= x^(sqrtx) 4. Integrate 1/xsqrt(x^2-4) dx 5.integrate sin^3 x. Dx