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# Integrals

### Integral domains and ideals

Let R be an integral domain and suppose that every prime ideal in R is principal. This exercise proves that every ideal of R is principal. (a) Assume that the set of ideals of R that are not principal is nonempty and prove that this set has a maximal element. [Use Zorn's Lemma.] (b) Let I be an ideal which is maximal with re

### Gradient Vector Fields and Line Integrals

Consider the vector field F = (x^2 + y^2, 8xy). Compute the line integrals and , where c1(t) = (t, t^2) and c2(t) =(t, t) for 0<=t<=1. Can you decide from your answers whether or not F is a gradient vector field? Why or why not?

Suppose that If f(0,0,0) = -5, find f(1, 1, 3). Hint: As a first step, define a path from (0,0,0) to (1, 1, 3) and compute a line integral. Please see the attached file for the fully formatted problems.

### Surface Integral of a Paraboloid of Revolution

Let S be the closed surface of the paraboloid of revolution z = ±(4 &#8722; x2 &#8722; y2 ) where &#8722;2 x, y +2. Evaluate the following surface integral directly and then by using the divergence theorem; where R is the position vector to a point on the surface and is the outward pointing normal at that point. See att

### Radial Force Fields and Work

Let F be the radial force field F= xi + yj. Find the work done by this force along the following two curves, both which go from (0, 0) to (5, 25). If C is the parabola... If C is the straight line segment... See attached file for full problem description.

### Convolution Integrals and Solving Differential Equations using LaPlace Transforms

1. Find the inverse of F(s) = s/[(s+1)(s^2+4)] the answer can be left as a convolution integral 2.Solve y''+y'+(5/4)y=y(t) y(0)=y'(0)=0 using LaPlace Transforms sin(t) t greater or equal to 0 or less than Pi y(t) { 0 t greater or equal than Pi

### Triple Integrals : Volume of Region bounded by Spheres - Spherical Coordinates

Use spherical coordinates to evaluate the triple integral , where E is the region bounded by the spheres x^2+ y^2 + z^2 =4 and x^2+ y^2 + z^2 =4. Please see the attached file for the fully formatted problems. keywords: integrals, integration, integrate, integrated, integrating, double, triple, multiple

### Contour Integrals

Calculate integral using contour integration. Complete explanation is required. integral(-&#8734;->+&#8734;)dx/(1+x^2)^n+1 keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Contour Integrals

Calculate the integral using contour integration. Complete explanation is required. integral(o->pi/2)d(theta)/(a+sin^2(theta)) keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Contour Integrals

Calculate the integral using contour integration. Complete explanation is required integral(o->&#8734;)(logx)^2dx/(1+x^2) keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Contour Integrals

Calculate the integral using contour integration. Complete explanation is required integral(o->&#8734;) dx/(x^3+1) keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Taylor Polynomials and Partial Sums of Series

1 Write the Taylor polynomial with center zero and degree 4 for the function f(x) = e^-x 2 Determine the values of p for which the series &#8734; &#931; 1/(2p)&#8319; n=1 3 Calculate the sum of the first ten terms of the series, then estimate the error

### Applications of Integration : Finding Area Between Circles Using Polar Coordinates

Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x^2 + y^2 = 64 and x^2 - 8x + y^2 = 0. keywords: integration, integrates, integrals, integrating, double, triple, multiple

### error in numerical integration

See attached file for full problem description. Determine the values of n and h required to approximate the integral of xlnxdx in [1, 2] to within 10-5 and compute the approximation. a. Use composite Trapezoidal rule. b. Use the composite Simpson's rule. c. Use the composite Midpoint rule.

### Approximation of Two Integrals with Gaussian Quadrature

Approximate the following integrals using Gaussian quadrature with n = 3 and n = 4 then compare your results to the exact values of the integrals. See attached file for full problem description.

### Ring of Integral Hamilton Quaternions and Isomorphisms

Let I be the ring of integral Hamilton Quaternions and define N: I ->Z by N(a+bi+cj+dk) = a^2 +b^2 +c^2+d^2 a) Prove that N(k)= kk' for all k in I where if k=a+bi+cj+dk then k'=a-bi-cj-dk b)Prove that N(kr)=N(k) N(r) for all k,r in I c) Prove that an element of I is a unit iff it has norm +1. Show that I(with multiplicati

### How much work is done pulling all the chain to the top of the building?

See attached file for full problem description. A 100-ft length of steel chain weighing 15 lb/f is hanging from the top of a building. How much work is done in pulling all of the chain to the top of the building?

### Finding an Integral for the Positive Sense of a Circle

a) Compute the integral of xdz (|z|=r) for the positive sense of the circle in two ways first by using parametrization and second by observing that x=(1/2)(z+z conjugate)=(1/2)(z+r^2/z) on the circle. b) Compute the integral of dz/(z^2-1) (|z|=2) for the positive sense of the circle. PS - Here maybe we have to find first a p

### Calculus word problem: rates, volume, and time (determine the amount of water added)

(See attached file for full problem description) Water will be added to the city's reservoir tonight for a 4 hour period. The rate at which the water is added depends on the time and is given by the function dw/dt = 1000 + 100t 0 <= t <= 4 Where w is the volume in gallons and t is the time in hours. Determine the

### Exact area under curve

Find the exact area under the curve y = x2 + 3 from x = 1 to x = 4.

### Find the definite integral

Find the definite integral: &#8747;(3 + x^2)dx from x = 0 to x = 1.

### determine the cost function by integral

Determine the cost function C(x) that corresponds to the marginal cost and fixed cost given. Marginal cost = 30 - 0.05x, Fixed cost = \$100 (See attached file for full problem description)

### Integrals

Antidifferentiation &#8747;(e^x -6) dx

### Finding the Volume of a Solid of Revolution

Problem: The region R is bounded by the graphs of x - 2y = 3 and x = y2. Find the integral that gives the volume of the solid obtained by rotating R around the line x = -1. I'm having a hard time setting up the integral, I think that I have the concept for finding the area of a 2d object using an integral but can't figure out

### Finding Integrals (8 Problems)

(See attached file for full problem description with proper symbols) --- Answers and working for Integration questions: 1.Integrate the following functions with respect to &#61553;. (i) sin(5&#61553; - 4) (ii) cos(3 - 2&#61553;) 2. Integrate the following functions with respect to x. (i) 4e-3x (ii) (

### Find an upper and lower bound for the integral using the comparison properties of integrals.

Find an upper and lower bound for the integral using the comparison properties of integrals. My Work. (I'm pretty sure I've made an Error) Integral lies between 0.5 and 1.0 (this is wrong though since it's .40)

### Applications of Integrals : Velocity and Acceleration

Newton discovered that the falling acceleration of all objects in a vacuum, regardless of their sizes and weights, is the same. A raindrop falls down to earth with the same acceleration as a big metal ball drops from the edge of a building. He came up with the value of 9.8 meters per square second (s2) for the falling accelerati

### Applications of Integration: Calculating Work

How much work is needed in exercise 15 to pump 4ft of water out over the top (a) when the tank is full? when the tank has 4ft of water in it? Exercise 15 A conical tank (inverted right circular cone) filled with water is 10ft across the top and 12ft high. How much work is needed to pump all the water out over the top?

### Integration, limits, and curves

Note: x is used as a letter only not as a multiply sign 1. Find the volume of the solid generated by revolving the region enclosed by y= x^(1/2), y=0, x=4 about the line x=6. 2. Find the arc length of the graph of the function y = x^(3/2) - 1 over the interval [0,4] 3. Integrate &#8747; [(Pi / 2) / 0] x cos x dx

### Cartesian Coordinates, Convergence and Divergence

1. Find the equation of the tangent line in Cartesian coordinates of the curve given in polor coordinates by r = 3 - 2 cos Ø, at Ø= (&#960; / 3) 2.Test for convergence or divergence, absolute or conditional. If the series converges and it is possible to find the sum, then do so. a) &#8721;[&#8734;/n=1] (3/ 2^n)