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# Integrals

### Riemann integral

Define f(x) = x if x is rational and f(x)=0 if x is irrational. Compute the integration 1top0bottom f dx and integration 1top0bottom f dx. is f integrable on [0,1]? the first integration is upper sums and the second is lower sums

### Calculating Double Integral in Two Ways

See attached. Let s be the portion of the unit sphere above the xy plane. Calculate the double integral of z.da over S in two ways: 1. Directly as the integral of a function over a surface. 2. By noting that z.da = (0,0,0).(x,y,z) = curlF.nda where the field F is F(x,y,z) = (-y/2,x/2,0) and S is gven in an upward orientat

### Isomorphism Subjective Subsets

1. Determine (up to isomorphism) all semisimple rings of order 1008. How many of them are commutative? (Recall that finite division rings are fields.) 2. A ring R is Boolean if x2 = x for all x  R. In a Boolean ring R, show that a. 2x = 0 for all x  R. b. R is commutative. c. Every prime ideal P is maximal and R/

### The Divergence Theorem

Please see the attached file for the fully formatted problem. A spherical surface K is given by the equation x^2+y^2+z^2=25 Let S be part of the spherical surface above the plane z=4 calculate the double plane integral over S of F.n where F = y^2-xz, x^2-yz, z^2-xy and n is teh unit vector to the surface S with positiv

### Integration and length

Assume that length of the fish increases according to the law: dL/dt = 2.45e^(-0.1t) where length is in cm and t in weeks.Assume that at time t = 0 the is is being born with length 0.5. When will a fish have length 14 cm?

### Multiple Integrals and vector calculus

Multiple Integrals and vector calculus. 1. To find the mass of a thin-shelled ellipsoid, you have to: A) Determine the inside surface area and then multiply by the shell's thickness B) Determine the average of the inside and outside surface areas and then multiply by the shell's thickness C) Find the difference

### Triple integral

Use a triple integral to find the volume of the given solids. 1.)the tetrahedron bounded by the coordinate planes and the plane 2x + 3y + 6z = 12 2.)the solid enclosed by the parabolas z = x^2 + y^2 and z= 0 and x + z = 1

### Curve integral and radius vector

Please see attachment as the task is hard to write in pure text. Brainmass doesn't support Latex in 'Post-a-problem' right? Anyway, short description: I have this curve C defined by x=sin(2t), y=1-cos(2t), z=2cos(t) where t lies between (or equal to) -pi and pi. - How to define a radius vector from origo to a point P at t

### Double Integral Vectors

Please see attachment, I wrote the task in latex, previewed it and took a screen shot. Task is as follows: A spherical surface K is given by the equation x^2 + y^2 + z^2 = 25. Let S be the part of the spherical surface that is above the plane z=4. Calculate the plane integral int(int(vect'F' * vect'n' dS)) defined by S.

### Surface integral of a vector field.

Please see the attached file for the fully formatted problems. The flow rate of a fluid is described by vector F in units of kg/m^2/s It is flowing through a membrane whoes surface S is described by: S = x^2+y^2+z^2=1 z>sqrt(3/4) The vector field is defined as: F(r) = [0,0,2-x^2-y^2+z] Calculate the rate of mass

### Brownian Motion

Please provide a detailed solution to the attached file. We are interested in estimating the response function of a Neuron...

### Green's Theorem and Line Integrals

Let gamma1 be given by (x,y)=(cos t , sin t), 0<t<pi/2 and gamma2 be given by (x,y) = (1-u,u) 0<u<1 compute: Integral of fdx + gdy over gamma1 Integral of fdx + gdy over gamma2 where f(x,y) = xy and g(x,y)=x+1 see file for more details

### Integral domain

Let R be an integral domain. An R-module M is called divisible if for every x in M and every nonzero r in R, there exists y in M such that ry=x...(see attached)

### Applications of Integration: Word Problems & Newton's Method

10 pts. 2. A farmer with 750 feet of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible area of the four pens? 10 pts. 3. Use Newton's method to find the approximations from x1 to x3 to find the third approximation of t

### Applications of Integrals : Fluid Force on a Plate

2. A triangular shaped plate with base of 12 ft. And height of 16 ft. Is submerged vertically in the water so that its base is 20 ft. below water. A. Sketch the plate on an appropriate coordinate system. B. Determine the fluid force on the plate.

### Application of integration

1. A cylindrical tank of diameter 5ft and height 10 ft is full of water. The water is pumped out over the top. Find the work required to a. Empty the entire tank. B. Empty half of the tank.

### Evaluation of the definite integral

The problem is read, set up and evaluate the definite integral for the area of the surface generated by revolving the curve. y=(x^4/8)+(1/4x^2),1 is less than or equal to t and t is less than or equal to 3, about the x axis. also there's a part b which reads.x=1-t^2,y=2t, 0 is less than or equal to t and t is less than or equal

### Principal and Prime Ideals

Let R be a commutative ring with 1. Show that the principal ideal generated by x in the polynomial ring R[x] is a prime ideal if and only if R is an integral domain. Prove that (x) is a maximal ideal if and only if R is a &#64257;eld. I have a solution for this problem but it starts with:First, note that a polynomial f in R[

### Ideals

Let R be a commutative ring with 1. Prove that the principal ideal generated by x in the polynomial ring R[x] is a prime ideal if and only if R is an integral domain. Prove that (x) is a maximal ideal if and only f R is a field.

### Setup and evaluation of integral over region of xy-plane

Sketch the region defined by x >= 0; x^2 + y^2 <= 2 and x^2 + y^2 >= 1. Write down the integral of the function f(x, y) = x^2 over the region in each of the two possible orders, and evaluate both integrals.

### Mathematics - Integration Applied to Commerce

The monthly sales at an import store are currently \$10,000 but are expected to be declining at the rate of S'(t) = -10t^2/5 dollars per month t months from now. The store is profitable as long as the sales level is above \$8,000 per month. a) FInd a formula for the expected slaes in t months. b) What sales figure shou

### Integration of Residents in a Town

The administrators of a town estimate that the fraction of people who will still be residing in the town t years after they arrive is given by the function f(t) = e^ -0.04t. If the current population is 20,000 people and new townspeople arrive at the rate of 500 per year, what will be the population 10 years from now?

### Integration applied to Commerce

Suppose that t years from now, one investment plan will be generating profit at the rate of P'1(t) = 100 + t^2 hundred dollars per year, while a second investment will be generating profit at the rate of P'2(t) = 200 + 2t hundred dollars per year. a) For how many years does the rate of profitability of the second investmen

### Mathematics - Calculus - Integration in Commerce

Profit over the useful life of a machine - Suppose that when it is t years old, a particular industrial machine generates revenue at the rate R'(t) = 6,025 - 8t^2 dollars per year and that operating and servicing costs accumulate at the rate C'(t) = 4,681 + 13t^2 dollars per year. a) How many years pass before the pro

### Differential Equations Formatting

See attached for formatting 1. Considering the differential equation y' = (y/x)3 : a. Discuss existence and uniqueness of solutions. b. Determine if exist constant solutions. c. Determine the general integral. d. Solve Cauchy problems y(3) = -1, y(3) = 0 and determine the maximal interval of solutions. 2) Integrate the

### It is estimated that x years from now, the value V(x) of an acre of farmland ...

It is estimated that x years from now, the value V(x) of an acre of farmland will be increasing at the rate of V'(x) = 0.4x^3 / square root of 0.2x^4 + 8,000 dollars per year. The land is currently worth \$500 per acre. a) FInd V(x) b) How much will the land be worth in 10 years?

### Price of Chicken Forecast

In a certain section of the country, it is estimated that t weeks from now, the price of chicken will be increasing at the rate of p'(t) = 3 * square root of t + 1 cents per kilogram per week. If chicken currently costs \$3 per kilogram, what will it cost 8 weeks from now?

### Mathematics - Calculus - Integration.

A study indicates that x months from now, the population of a certain town will be increasing at the rate of 10 + 2 sqrt x people per month. By how much will the population increase over the next 9 months?

### Riemann-Stieltjes integral

Let a(x)=x^3. From the definition, approximate the Riemann-Stieltjes integral....for each of the following partitions

### An Explanation for a Line Integral

Please address the following question: Show that the line integral is independent of path and evaluate the integral. Integral_c (2y^2 - 12x^3y^3)dx + (4xy - 9x^4y^2)dy C is any path from (1, 1) to (3, 2)