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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

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