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# Integrals

### 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

### How to Find the Definite Integral

View attachment for the full problem.

### Differential Equation Formatted Problems

Please see the attached file for the fully formatted problems. y(t) + S t-->0 (t-tau)y(tau) dtau = e^t

### Integration by Parts Primitive Function

Determine the indefinite integral: z = (Integral Sign or "Long S")20xe^-4x dx. I got z = 5xe^-4x + 5/4 e^-4x + C - Does that seem right?

### Simpson's Rule Integrals

Find the integral of e^-x dx from x = 0 to 1 with Simpson's rule using 10 strips.

### Equations using Adding and/or Subtracting

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

### Evaluate the Surface Integral

Please see the attached file for the fully formatted problems. Evaluate the surface integral SSs x dS, where S is part of the plane x = 2y = 3z = 6 that lies in the first octant.

### Green's Theorem : Evaluating a Line Integral

Please see the attached file for the fully formatted problems. Use Green's Theorem to evaluate the line integral Sc xy dx +x^2y^3 dy where C is the triangle with vertices (0,0), (2,0) and (2,2).

### Gradients : Find a Function and Evaluate an Integral

Please see the attached file for the fully formatted problems. (a) Find a function f so that grad(f) = yi + (x + 3y^2)j (b) Use part (a) to evaluate Sc grad(f) dt where C is the path starting at (0,2) goes down the y-axis to (0,0), along the x-axis to (2,0).

### Iterated Integral Evaluated

Please see the attached file for the fully formatted problems. Evaluate the iterated integral S 1-->0 S z--> 0 S x+z --> 0 x dydxdz

### Polar Coordinate Solution

Please see the attached file for the fully formatted problem. By changing to polar coordinates, evaluate SSR (x+y) dA where R is the disk of radius 4 centered at the origin.

### Integral Linear Operator Functions

L[y](x)= Integral between 0 and 1 (x-t)^2 y dt is a linear operator. or See attachment...

### Representation of Integration Powers Implied

How do you integrate: [a^x exp(-a) ] / x ! (The ^ represents 'to the power of ' so a^x implies a raised to the power x)

### Green's Function for a Differential Equation

Problem attached. Over the interval infinity -infinity < x < infinity consider

### 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?

### Combinatorial Result about the Binomial Coefficients

Please see the attached PDF file for the fully formatted problem. By the Binomial Theorem .... Therefore .... Evaluate the integral to get .... by a similar line of reasoning. Since this is an analysis problem, please be sure to be rigorous, and include as much detail as possible so that I can understand. Please also

### Evaluate an Integral

Please see the attached file for the fully formatted problems. Reverse the order of integration to evaluate S 2 ---> 0 S 1 ---> y/2 cos(x^2) dxdy

### Compute the volume of a solid.

A solid has a rectangular base in the x,y-plane with sides parallel to the coordinate axes and with opposite comers at points (0,0) and (4,6). The solid's sides are vertical, and the top is determined by the plane z = 3x + y. Compute the volume of the solid.

### Definite Integral : Area of a Region

Find the area of the region {(x, y)|0 <_ y <_ x^3 &#8722; 12x + 6, 4 <_ x <_ 6} <_ is to be taken as less then or equal to

### Evaluating an Integral With a 2nd Order Pole using the Residue Theorem

Evaluate the integral from 0 to INF of: (x^a)/(x^2 +4)^2 dx, -1 < a < 3 We are to use f(z)= (z^a)/(z^2 +4)^2, with z^a = e^(a Log z), Log z= ln|z| + i Arg z, and -pi/2 < Arg z < 3pi/2. I have found the residue at 2i to be: [2^a(1-a)/16]*[cos ((pi*a)/2) + i sin ((pi*a)/2). Please let me know if this is correct and

### Evaluating an Integral using Jordan's Lemma...

The problem is: Evaluate the integral from 0 to INF of: [(x^(1/3))*(ln x)]/(x^2 +9) dx by using f(z)= [(z^(1/3))*(Log z)]/(z^2 +9), with -pi/2 < Log z < 3pi/2. Also, with z^(1/3)= e^[(1/3)Log z]. We are to use the curve C: from -R to -p, -p to p around origin, p to R, and Cr from 0 to pi. Many thanks in advance

### Evaluating an Integral using Jordan's Lemma

The problem is to find the value of the integral from 0 to INF of [(ln x)^2]/(x^2 +9). We are to use f(z)= [(Log z)^2]/(z^2 +9), where -pi/2 < Log z < 3pi/2. We are to use the curve C from -R to -p along the real axis, -p to p around 0, p to R along the real axis, and the curve Cr from 0 to pi. I am having several probl

### Non Homogenous First Order Differential equation.

Solve the differential equation. (x^2+1)dy/dx + xy = x

### Indefinite Integral : Trigonomeric Substitution

Please see the attached file for the fully formatted problem. Integrate: S x^2/sqrt(25 - x^2) dx

### Find the Limit of the Improper Integral

Find the limit of the improper integral on attached file.

### Definite Integral Expressions

Please see the attached file for the fully formatted problem. Integrate : S x sin 2x dx pi--> 0

### Area Under a Curve

Please see the attached file for the fully formatted problems. Find the area under the curve from x = 0 to x = 2 of y = ½ x^2 + 1

### 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?

### Find the indefinite integral

Find the indefinite integral of x divided by the square root of x - 1