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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 ≤ 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

Business statistics and calculus

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

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.

Revenue Function and Definite Integral of Revenue Function

Revenue at day D = (200 + 10D - 100P)*P D refers to the day, with Monday being 1, Tuesday 2, etc up to Friday with a value of 5. The assignment is as follows: 1. If you are charging $1 per cup, what is your revenue for each of the five days? What is your total revenue for the week? 2. What is the indefinite integral

Stokes Theorem, Curl and Positively Oriented Hemisphere

7) Use Stoke's Theorem to evaluate curl F*dS S is the hemisphere oriented in the direction of the positive x-axis. 8) Use Stokes Theorem to evaluate C is the boundary of the part of the plane 2x + y + 2z = 2 in the first octant. 9) Suppose that f(x,y,z)= , where g is a function of one variable such that g(2) = -

Green's Theorem, Positively Oriented Curve, Ellipse

Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. ∫c xdx + ydy. C consists of the line segments from (0,1) to (0,0)...and the parabola y = 1 -x^2.... Use Green's theorem to evaluate the line intgral along the positively oriented curve. ∫c sin y dx + x cos y dy

Integration Techniques and Applications

1. Find the indefinite integrals for the following functions: a. f(X) = 10000 b. f(X) = 20X c. f(X) = 1- X2 d. f(X) = 5X + X-1 e. f(X) = 12- 2X f. f(X) = X3 + X4 g. f(X) = 200X - X2 + X100 2. Find the definite integral for the following functions: a. f(X) = 67 over the interval [0,1] b.

Definite Integral as the Limit of a sum BrainMass Expert explains

Calculus Integral Calculus(I) Definite Integral as the Limit of a sum Method of Summation Definite Integral It is an explanation for finding the integral by using the method of summation(part I). Find by the method of summation the value of: (a) ∫ e^( - x)dx, where the lower limit is a and the upper limit is b

Congruences, Primitive Roots, Indices and Table of Indices

6. Let g be a primitive root of m. An index of a number a to the base (written ing a) is a number + such that g+&#8801;a(mod m). Given that g is a primitive root modulo m, prove the following... 7. Construct a table of indices of all integers from.... 8. Solve the congruence 9x&#8801;11(mod 17) using the table in 7. 9.

Helicoid Integral

Evaluate &#8747;&#8747;S &#8730;(1 + x^2 + y^2) dS where S is the helicoid: r(u,v) = u cos(v)i + u sin(v) j + vk , with 0 &#8804; u &#8804; 3, 0 &#8804; v &#8804; 2pi. Please see the attached file for the fully formatted problem.

Flux Integrals

Suppose is a radial force field,... is a sphere of radius...centered at the origin, and the flux integral.... Let be a sphere of radius... centered at the origin, and consider the flux integral... . (A) If the magnitude of... is inversely proportional to the square of the distance from the origin,what is the value of..

Newton's Law of Cooling : Average Temperature

If a cup of coffee has temperature 95 degrees Celsius in a room where the temperature is 20 degrees Celsius, then according to Newton's Law of Cooling, the temperature of the coffee after t minutes is T(t)=20+75e^(-t/50). What is the average temperature of the coffee during the first half hour?

Lebesgue Integral and Monotone Convergence Theorem

Let f be a nonnegative integrable function. Show that the function F defined by F(x)= Integral[from -inf to x of f] is continuous by using the Monotone Convergence Theorem. From Royden's Real Analysis Text, chapter 4. See the attached file.

Solve Sequences Used in Given Situation

There is a rope that stretches from the top of Maidwell building to a tree on the racecourse, and the length of this rope is 1km. A worm begins to travel along the rope at the rate of 1cm each second in an attempt to get to the other end. then a strange thing happens... some malevolent deity intervenes to make life even hard

Evaluating integrals

Evaluate the following integrals: (1)The integral of (6sin[2x])/sin(x)dx=____+C (2)The integral of (7-x)(3+[x^2])dx=____+C (3)The integral from 3 to 7 of ([t^6]-[t^2])/(t^4)dt=____+C (4)The integral of (6sin[x])/(1-sin^2[x])dx=____+C

Definite and indefinite integrals

Thank you in advance for your help. Evaluate the following definite and indefinite integrals: (1)The integral of (2x)/([x^2]+1)dx (2)The integral of [(arctan(x))/([x^2]+1)]dx (3)The integral of sqrt([x^3]+[1x^5])dx (4)The integral of (2+x)/([x^2]+1)dx (5)The integral of (2x)/([x^4]+1)dx (6)The integral from 0 to 2 of (x

Proving an Integral Equation

If f is continuous for all real numbers, prove that the integral from a to b of f(x+c)dx=the integral from a+c to b+c of f(x)dx. For the case where f(x) is greater than or equal to 0, draw a diagram to interpret this equation geometrically as an equality of areas.

Integrals : Rate of Change Word Problem

Water flows from the bottom of a storage tank at a rate of r(t)=200-4t liters per minute, where 0 is less than or equal to t and t is less than or equal to 50. Find the amount of water that flows from the tank during the first 10 minutes.