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# Integrals

### Integration by Substituton

I am having problems understanding how to solve linear equations. For example, dy/dt = -4y + 3e^-t Can you solve this step-by-step so that maybe I can understand how to do it myself?

### Explicit Rule & Recursive Rule & Integrals & Exponential Growth

1.) For the arithmetic sequence {-10,-2,6,14,}, find: a.) a recursive rule for the nth term b.) an explicit rule for the nth term 2.) For the geometric sequence {512,256,128,64,...}, find: a.) a recursive rule for the nth term b.) an explicit rule for the nth term 3.) Determine whether each of the follo

### Vector Fields, Fundamental Theorem of Line Integrals

1. Find the curl of the vector field F at the indicated point: 2. Evaluate the following line integral using the Fundamental theorem of line Integrals: 3. Use Green's Theorem to calculate the work done by the force F in moving a particle around the closed path C: 4. Find the area of the surface over the part of the

### Solid of Revolution

Find the volume of the solid bounded by x = 0, y = 0, z = 0 and x + 2y + 3z = 6 by triple integration. keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Integrals : Rayleigh's Energy Theorem ( Parseval's Theorem )

Evaluate the following integrals using Rayleigh's energy theorem (This is Parseval's theorem for Fourier transforms). All integrals spans between -&#8734; and &#8734;. I1 = &#8747; df / [&#945;2 + (2&#960;f)2] I2 = &#8747; sinc2(&#964;f) df I3 = &#8747; df / [&#945;2 + (2&#960;f)2]2 I4 = &#8747; sinc4(&#964;f) df

### Center of Mass of a Lamina; Polar Coordinates and Jacobians

Find the mass and center of mass of the lamina bounded by the graphs of the given equation and of the specified density. Use polar coordinates to evaluate where R is the region enclosed by . Suppose the transformation T is defined by and . Find the Jacobian, , of T. Please see the attached file for t

### Lebesgue Integration

For what values of a in R (real numbers) is the function (1+x^2)^a in L^2 NOTE: let L^2 be like in Lebesgue integration where the set of all measurable functions that are square-integrable forms a Hilbert space, the so-called L2 space. keywords: integration, integrates, integrals, integrating, double, triple, multiple, r

### Volume of Solids of Revolution

Please help with the following problems. Please provide step by step calculations and diagrams. For the following problem, sketch the region and then find the volume of the solid where the base is the given region and which has the property that each cross-section perpendicular to the x-axis is a semicircle. 1) the regio

### Creating a Function Command in SQL

Evaluate the integral of several trigonometric functions.

### Determining where a function is continuous, Epsilon-Delta Continuity Problem

Determine where the function f(x) = x + [|x^2|] - [|x|] is continuous I don't understand how to work this problem. Can someone show and explain how to solve this problem in detail? I've asked for help on this problem before but I still don't understand it. The function is continuous everywhere but 0. I don't understan

### Definite Integrals

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Indefinite Integrals

Please see the attached file for the fully formatted problems. keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Definite Integrals

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple, areas keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Find the indefinite integrals and check the results by differentiation.

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Indefinite Integrals

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Set up the definite integral that gives the area of the region.

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Integrals

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Integrals

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Trigonometric Integrals

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Integrals

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Integrals

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, evaluating, calculate, calculating, prove, proving

### Evaluate the given line integral and the given surface integral.

Do the following: (1) Evaluate Int(P(x, y) dx + Q(x, y) dy) over the curve C, where P(x, y) = y^2, Q(x, y) = 3x, and C is the portion of the graph of the function y = 3x^2 from (-1, 3) to (2, 12). Here, "Int" stands for integral. (2) Use the Divergence Theorem to evaluate the surface integral Int(F*n ds) over the surface S

### Double Integration

See attached file for full problem description.

### Area and volume using multiple integrals

1 Find the area of the part of the surface 2z = x^2 that lies directly above the triangle in the xy-plane with the vertices at (0,0),(1,0) and (1,1). 2 Find the volume of the region in the first octant that is bounded by the hyperbolic cylinders xy = 1, xy = 9, xz = 4, yz = 1, and yz = 16. Use the transformation u = xy, v =

### Finding the Volume of Solid of Revolution using the Disk Method

Find the volume of the solid formed by revolving the region bounded by y=x^3, x=2, and y=1 about the y-axis. keywords: integrals, integration, integrate, integrated, integrating, double, triple, multiple

### Stokes' Theorem : Curls and Surface Integrals

Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

### Stokes Theorem to Calculate the Surface Integral of the Curl

See attached file for full problem description. Use Stokes' theorem to evaluate the surface integral of the curl: where the vector field F(x,y,z) = -12yzi + 12xzj + 18(x^2+y^2)zk and S is the part of the paraboloid z = x^2 + y^2 that lies inside the cylinder x^2 + y^2 =1, oriented upward.

### Integration Word Problems : Rate of change

Suppose that a tank initially contains 2000 gal of water and the rate of change of its volume after the tank drains for t min is '(t)=(0.5)t)-30 (in gallons per minute). How much water does the tank contain after it has been draining for 25 minutes? keywords: integration, integrates, integrals, integrating, double, triple, m

### Volume of a solid

The region R is bounded by the graphs of x-2y = 3 and x=y^2. Set up (but not evaluate) the integral that gives the volume of the solid obtained by rotating R around the line x=-1.

### Unique Factorization Domain with Quotient Field

Let R be an integral domain with quotient field F and let p (X) be a monic polynomial in R[X] : Assume that p (X) = a (X) b (X) where a (X) and b (X) are monic polynomials in F [X] of smaller degree than p (X) : Prove that if a (X)is not in R[X] then R is not a UFD(unique factorization domain). Deduce that Z[2sqrt2] is not a