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# Integrals

### method of summation the value of the integral is sin(x) and cos

Integration as the limit of a sum (II) Find by the method of summation the value of: a) The integral (from 0 to 1/2*pi) of sin(x). b) The integral (from a to

### Mathematica integration

I am new to Mathematica and did derive the answer to the following but I can not get the information as to the steps taken to derive it. I am using the trapezoidal and Simpson's rules to evaluate S20 x2 dx ( the S should be the variant symbol) Compare with exact value. My answer is 8/3 but I can not get Mathema

### Z-Transforms & Grad, Div, Curl

Please see the attached file for the fully formatted problems.

### Volume of a solid..

Please see word attachment for clearer view of the problem. Volume: Find the volume of the solid generated by revolving the region bounded by the graphs of y = xe^-x, y = 0, and x = 0 about the x-axis.

### Calculus Functions to Evaluate an Integral

Please help with various Calculus questions. You do not need to show your work for this one because I would simply like to compare your answers with mine so that I am sure that I did everything correct on mine. Please just write your exact answer after each number. I will know which problems I will have to study in detail w

### Partial Derivative and Double Integral

The problems are attached 1 -5 based on Chapter Partial Derivative - (Maximum & Minimum Values and Lagrange Multipliers 1. Locate all relative maxima, relative minima, and saddle points of the surface defined by the following function. 2. Consider the minimization of subject to the constraint of (a) Draw the

### Evaluate the integral

The following expression describes the total electric current to pass in the circuit please see attached

### Integral domain

There is integral domain with exactly six elements. Disprove or Prove

### Acceleration and Velocity

An object is thrown downward from the top of a building with an initial velocity of 30 m/s. Assuming a positive direction of y measured downward from the top, derive an expression for (a) the velocity and (b) the displacement as a function of time. Assume y(0)=0

### MATLAB Programming

Develop a program (M-File) called 'integrate' that will perform a first-order numerical approximation, yi(t), of the running integral with respect to time of an array of experimental data y(t). The M-File must also perform another first order approximation, yi2(t), of the first integral resulting in a double integration of the

### Equations using Adding and/or Subtracting

1. Solve: x + 79 = 194 2. Solve: x - 56 = 604.

### Area of a property using integration

I've included the problem as a JPEG . Thank you You own a plot of riverfront property which is pictured in the figure. Your property runs along the x-axis from x=0 to x=100 and is bounded by the lines x=0, x=100 and the River Sine. 1. What is the equation of the River Sine? 2. What is the area of the plot? 3. You have \$6

### Mean Value Theorem for Harmonic Functions : Green's Identity

Please see the attached file for the fully formatted problems. Let h 2 C2(R3) be harmonic (h = 0). Use Green's identity for .... to show that ...is independent of the value of R. Then deduce the mean value theorem .... Now what can you say if limx!1 h(x) = 0?

### Alberti Cipher Disk and Enigma Cipher

Why did the Alberti Cipher disk have numbers on it? Describe how the disk was used. What made it secure? What was the impact of this disk on cryptology? How does enciphering and deciphering differ on the Enigma? What weakness in the Enigma did the Poles use to break Enigma ciphers?

### Relating Transform of a Function and Transform of the Derivative

Please see the attached file for the fully formatted problems. Problem statement: What really makes Laplace transforms work for differential equations is the relationship between the transform of a function and the transform of the derivative of that function. Therefore, the formula you will prove below is key to all that

### Volume of Space Formed by Intersection of Three Cylinders at Right Angles

The figure shows a solid enclosed by three circular cylinders with the same diameter that intersect at right angles.... see attachment for figure and remainder of question. PART 2 ONLY!

### Integral Domains and Fields: Embedding Theorem

Problem: Note: C is set containment If R is an integral domain, show that the field of quotients Q in the Embedding Theorem is the smallest field containing R in the following sense: If R C F, where F is a field, show that F has a sub-field K such that R C K and K is isomorphic to Q.

### Real Analysis: Multivariate Probability Distributions

Please see the attached file for the fully formatted problems. Multivariate Probability Distributions f(y1,y2) = 3y1, 0<y2<y1<1 f(y1,y2) = 0, elsewhere Find P(0<y1<0.5, 0.25<y2)

### Integral : Trigonometric Substitution

Int sqrt{1+x^2}/x dx

### Integration By Trigonometric Substitution: Solving Trickier Problems?

Hello! I'm having trouble using Trigonometric Substitution to find the anti-derivative of non-simple integrands. For details on my situation, please consult my missive, which I've included as an attachment in MS Word '95 (WordPad compatible) and Adobe PDF (ver 3+) files. (The files contain identical information; if you can re

### Multivariable Calculus : Double Integral - Polar Coordinates

( &#61682; ^n_r means that n is on the top of the &#61682; and r is on the bottom) Evaluate the given integral by first converting to polar coordinates: &#61682; ^1_0 &#61682; ^(square root of 1 - x^2)_0 (1/(square root of 4 - x^2 - y^2)) dy dx &#61682;: is the integral symbol

### Differential Equation

Given dy/dx= -xy/(ln y), where y>0 find the general solution of the differential equation What solution satisfies the condition that y=e^2 when x=0... express in y=f(x) Why is x=2 not in the domain found from that?

### Integrals and Continuity Calculus

20) If the function f is continuous for all real numbers and lim as h approaches 0 of f(a+h) - f(a)/ h = 7 then which statement is true? a) f(a) = 7 b) f is differentiable at x=a. c) f is differentiable for all real numbers. d) f is increasing for x>0. e) f is increasing for all real differentiable ans is B. Explain

### Double Integral

Can anyone please show me how to solve these double integrals, with a step by step approach. I know the answer is 63 - but Ive tried so many times & I don't know where I'm going wrong. upper limits are 1&y=2 x+4y^2 dydx + lower limits are -2&y=-x upper limits are 4 & y=2 x+4y^2 dydx lower limits ar

### Double Integral over a Square Region

Use a transformation to evaluate the double integral of f(x,y) given by f(x,y)=cos(2x-y)sin(x+2y) over the square region with vertices at (0,0) (1,-2) (3,-1) & (2,1) (My notes from class-uses substitution, change of variables) I have let u=(2x-y) & v=(x+2y) using substitution (change of variables)

### Population model

In some populations, the amount of births is directly proportional to the population at any given point in time and the amount of deaths is directly proportional to the square of the population at any given point in time. 1. Write an equation that models the change in a population that fits the above description. Make sure t

### Find the Laplace Transform of the function F(t) = (1 - e^(-at))/a. Prove by the method of contour integration that F(t) is itself the Laplace Transform of the function arrived at.

Laplace Transform Application of Complex Inversion Integral Formula (Bromwich's Integral Formula) Problem:- Find the Laplace Transform of the function F(t) = (1 - e^(

### Partial Fraction Decomposition

Please see the attached file for the fully formatted problems. Partial fraction decomposition is a technique used to convert a fraction with a polynomial numerator and a polynomial denominator into the sum of two or more simpler fractions. It eases integration by reducing it to the sum of integrals, each of which will most l

### Volume of Revolution

1. The shaded region R, is bounded by the graph of y = x^2 and the line y = 4. a) Find the area of R. b) Find the volume of the solid generated by revolving R about the x-axis. c) There exists a number k, k>4, such that when R is revolved about the line y = k, the resulting solid has the same volume as the solid in par

### Find the Laplace Transform of the function F(t) = (1 - e^(-at))/a. Prove by the method of contour integration that F(t) is itself the Laplace Transform of the function arrived at. It is an Application of Complex Inversion Integral formula (Bromwich's Integral Formula) for finding the Laplace Transform of functions.

Laplace Transform Application of Complex Inversion Integral Formula (Bromwich's Integral Formula) Problem:- Find the Laplace Transform of the function F(t) = (1 - e^(-at))/a