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# Integrals

### Complex integration using series expansion of analytic functions

I want to check my answer: Evaluate the following integrals: integral over gamma for (sin z)/z dz, given that gamma(t) = e^(it) , 0=<t=<2pi ( e here is the exponential function) My work: sin z = z - z^3/3! + z^5/5! + ... + (-1)^n (z^(2n-1))/(2n-1)! + ... divide by z we get (sin z)/z = 1 - z^2/3! + z^4/5!

### Complex Integration using power series expansion of analytic functions

Evaluate the following integrals: (a) integral over gamma of (e^z - e^-z)/(z^n) dz, where n is positive integer and gamma(t) = e^(it), 0 =< t =< 2 pi (b) integral over gamma of (dz/(z^2 + 1) ) where gamma(t) = 2e^(it), 0 =< t =< 2pi ( Hint: expand (z^2 + 1)^-1 by means of partial fractions PLEASE USE POWER SERIE

### Triple Integral on Specific Solid

Compute a triple integral over a specific solid that would be very difficult in rectangular coordinates, but easy in parabolic cylindrical coordinates, u, v, z, where x=(1/2)(u^2-v^2) y=uv z=z You must come up with the solid. Remember that the Jacobian determinant (u^2+v^2) must be used when transforming an integral to this

### Complex Integration and Polynomials : If gamma(t) = Re^(it), 0 =< t =< 2 pi, show that the integral over gamma p'(z)/p(z) dz = 2 ( pi ) i n.

Let P(z) be polynomial of degree n and let R>0 be sufficiently large so that p never vanishes in { z: |z| >= R}. If gamma(t) = Re^(it), 0 =< t =< 2 pi, show that the integral over gamma p'(z)/p(z) dz = 2 ( pi ) i n.

### Interpolate

(See attached file for full problem description with proper equations) --- 1. For medical purposes the level of sugar was measured in blood (in mg/dl). The samples were taken with 1/2 hour increments, as the following table shows: Initial sample 96 mg/dl After 30 min. 133 mg/dl After 60 min. 142 mg/dl After 90 min.

### Integration: Profit Function, Pesent Values and Future Values

An oil company discovered an oil reserve of 100 million barrels. For time t>0, in years, the company's extraction plan is a linear declining function of time as follows: q(t)=a-bt Where q(t) is the rate of extraction of oil in millions of barrels per year a time t and b= 0.1 and a =10 . a) How long does it take to exhaust

### 12 Problems - Integrals : Sum of Partial Fractions and Volume of Solid of Revolution

36)After t weeks, contributions in response to a local fund raising campaign were coming in at a rate of 2000te^-0.2t dollars/week. How much money was raised during the first five weeks. 38) Find the volume of the solid generated when the region under the curve y=sinx+cosx on the interval [0,pi/4] is revolved about the y axi

### Area Bounded by Curves & Volume of Solid of Revolution

Find the area of each polar region enclosed by f(theta) for a <=theta<=b 36) f(theta) = theta/pi, 0<=theta<=2pi PLEASE SHOW EVERY STEP IN SOLVING THESE-NO COMPUTER PROGRAMS PLEASE. 4) Identify each curve as cardiode, rose(state # of petals), leminscate, limacon, circle, line or none of above. a) r=2sin2theta b) r^2=2c

### Integrals Application Word problem : Supply and Demand Curves and Equilibrium

4. The demand curve for a product has equation p=20 e^(-0.002q) and the supply curve has equation p=0.02q + 1, where q is the quantity and p is the price in \$/unit. a) Which is higher the price at which 300 units are supplied or the price at which 300 units are demanded? Find both prices. b) Sketch the supply and deman

### Definite Integrals : Supply and Demand Curves and Equilibrium

6. The demand curve for a product has equation p = 100 e^(-0.008q) and the supply curve has equation p = (4&#8730;q) + 10 , where q is the quantity and p is the price in dollars/unit. a) At a price of \$50, what quantity are consumers willing to buy and what quantity are producers willing to supply? Will the market push price

### Definite Integrals Application Word Problem : f(t) = 100e^(-0.5t)

A service station orders a 100 cases of motor oil every 6 months. The number of cases of oil remaining t months after the order arrives is modeled by f(t) = 100e^(-0.5t) a) How many cases are there at the start of the six-month period? How many cases are left after the end of the six-month period? b) Find the average number

### Integrals, Marginal Cost Function and Marginal Profit

The marginal cost function of producing q mountain bikes is a) If the fixed cost in producing the bicycles is \$2000, find the total cost to produce 30 bicycles b) If the bikes are sold for \$200 each, what is the profit (or loss) on the first 30 bicycles c) Find the marginal profit on the 31st bicycle. Please see the atta

### Vector Calculus: Surface Integral

#12) Surface integrals; &#61682;s&#61682; G(r) dA. Evaluate these integrals for the given data. (show the details.) G=cosx + siny, S: the portion of x+y+z=1 in the first octant (See attached file for full problem description with equations) Question in Kreyszig's Advanced engineering mathmatics 8th ed.: section 9.

### Normed Space, Compactness and Transformation

Let X be a normed space, I closed interval ( or half-open on the right) and a = inf I, b = sup I. Let h : I -> [0,infinity) be a continuous function such that integral ( from a to b ) h(t)dt < positive infinity where integral from a to b represents the improper integral when I is not closed. Let epsilon > 0 and

### Measure Space and Bounded Integral Operator

If is a measure space and , show that defines a bounded integral operator. Please see the attached file for the fully formatted problems.

### Solving a Pfaffian equation for a complete integral

Hello. Thank you for taking the time to help me. I cannot use mathematical symbols, thus, * will denote a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. To simplify things, I will let p=u*x and q=u*y. Furthermore, I will use ^ to denote a power. For example, x^2 means x squared. Also,

### Green's function and Poisson's integral formula

Hello. Thank you for taking the time for helping me. The following is the problem which I need to solve (there are actually two parts): I need to construct Green's function for the Dirichlet problem (Laplace's equation) in the upper half plane R={(x,y) : y>0} and I must derive Poisson's integral formula for the half plane.

### Cauchy Integral Formula

(See attached file for full problem description and embedded formulae) --- Why can (1) be regarded as a special case of (2)? (1) Cauchy's Integral formula (no need to prove): is a simple closed positively oriented contour. If is analytic in some simply connected domain D containing and if is any point inside ,

### Problems with measurable functions

1) a. If g is measurable on [a, b] with g(x) 0 for all x [a, b], prove that 1/g is measurable on [a, b]. b. prove that every continuous real- valued function f on [a, b] is measurable. c. if f is differentiable on [a, b], prove that f' is measurable on [a, b]

### Vector Integrals : Stokes' Theorem and Vector Fields

7.. Given the vector field F(x,y,z) = xi + (x+2y+3z) j + z2 k Let C he the circle on the xy-plane, centered at the origin (0,0) and having as radius r=5. Let S be the part of the paraboloid z = 16? x2 ? y2 which lies above the xy-plane (z &#8804; 0). Use the Stokes's Theorem to evaluate the line integral of this vector field a

### Surface Integral Over a Portion of a Cone

Let S be the portion of the circular cone in a space that has an equation z^2= x^2 + y^2 and that lies between the planes z=7 and z=11. Given the scalar function...evaluate the surface integral over... (See attached file for full problem description)

### Jordan's Lemma and Loop Integrals

Without evaluating the improper integrals and find the numerical value q of their quotient by considering the loop integral where is the semi-circular loop indented at the origin. Explain why Jordan's Lemma (see below) is inadequate here, and write a complete formulation of a more general Jordan's lemma

### Static Moment: Symmetrical Trapezoidal Plate in a Liquid

A symmetrical trapezoidal plate has the following dimensions: The width of the parallel sides are, respectively, 2.5 and 4.5 ft. The perpendicular distance between those sides is 1.5 ft. The plate is submerged in a liquid in a vertical position with the parallel sides horizontal and the shorter parallel side at the tip and exact

### Heat Equation : Temperature Distribution on a Brass Rod

9. The temperature distribution u(x, t) in a 2-m long brass rod is governed by the problem ...... (a) Determine the solution for u(x, t). (b) Compute the temperature at the midpoint of the rod at the end of 1 hour. (c) Compute the time it will take for the temperature at that point to diminish to 5° C. (d) Compute the ti

Please see attached. Hi, I am having trouble doing these problems listed below. Please show me how to solve these problems for future reference. Thank you very much. I would like for you to show me all of your work/calculations and the correct answer to each problem. For Exercise 2, find the mode of the probability

### Riemann Integration, Partitions, Upper and Lower Sums

1. Suppose f: [a,b] &#61614;&#61522; is a function such that f(x)=0 for every x &#61646;(a,b]. a) Let &#61541; > 0. Choose n &#61646; &#61518; such that a + 1/n < b and |f(a)|/n <&#61541;. Let P ={a, a+1/n, b} &#61646; &#61520;([a,b]). Compute &#61525;(f,P) - &#61516;(f,P) and show that is less than &#61541;. b) Prove

### Advance Calculus Riemann Integration Concepts : Partitions, Upper and Lower Sums and Definition of the Riemann Integration

1. Suppose f: [a,b] &#61614;&#61522; is a function such that f(x)=0 for every x &#61646;(a,b]. a) Let &#61541; > 0. Choose n &#61646; &#61518; such that a + 1/n < b and |f(a)|/n <&#61541;. Let P ={a, a+1/n, b} &#61646; &#61520;([a,b]). Compute &#61525;(f,P) - &#61516;(f,P) and show that is less than &#61541;. b) Prove

### Computing areas and volumes using multiple integrals.

(1) Find the volume of the solid bounded by the paraboloid x2 + y2 = 2z, the plane z = 0 and the cylinder x2 + y2 = 9. (2) Find the volume of the region in the first octant bounded by x + 2y + 3z = 6. (3) Find the area of the solid that is bounded by the cylinders x2+z2 = r2 and y2+z2 = r2. (4) Find the volume enclosed by t

### Applications of the Change of Variables Theorem

(1) Find ... (x + y)2 dx dy...where R is the square with vertices (±1, 0) and (0,±1), (2) Let R now be the triangular region in the xy plane with vertices (1, 0), (2, 1), (3, 0). Find.... (3) Change the integral .... from rectangular to polar coordinates. See the attached file.