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    Integrals

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    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.

    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.

    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

    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?

    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

    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.

    Integral Domains, Fields and Subfields

    Problem: Note: Q is rational numbers, R is real numbers , sqrt() means square root Show that Q(sqrt(2)) is the smallest subfield of R that contains sqrt(2).

    Wave Packet: Orthonormal Functions and Complete Set

    1. Consider the set of functions ("wave packets") (see attached) where e is a fixed positive constant. a) SHOW that these wave packets are orthonormal. b) SHOW that these wave packets form a complete set.