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# Integrals

### Indented Path : Integral of Branch of Multiple-Valued Function

Show that... by integrating an appropriate branch of the multiple-valued function... over (a) the indented path in Fig. 97, Sec 75; (b) the closed contour in Fig. 99, Sec. 77. See attachment for equations and diagrams.

### Use a Function to Derive a Pair of Integration Functions

Use the attached function to derive the pair of integration functions {see attached for functions and diagram}

### Solving: Improper Integrals

Use residues to evaluate the improper integrals (see the attachment to view the problem).

### Evaluate the Following Integrals (4 Problems)

Evaluate the following integrals: 4 1. ∫ (2x-3)(x+2)dx 2 2 2. ∫ (2x+1)4dx 1 0.001 3. ∫ 50cos50πt dt -0.001 π/2 4. ∫ 5sin(2t-(π/6)dt 0

### Integral

Find the integral of [x/(-x^2+6x+13)^(1/2)]dx

### Find an integral by substitution.

Find the integral of dx/1+(x)^1/2

### Integration by Parts

Integral of (t^2)Ln(t-2)dt

### Polar Coordinates : Evaluating an Improper Integral

A) Using polar coordinates, evaluate the improper integral {see attached} B) Use part A to evaluate the improper integral {see attachment} *Part A is completed, but I need some help with Part B

### Residues and Poles : Value of Integral (Counterclockwise around a Circle)

Fine the value of the integral {see attachment} taken counterclockwise around the circle: (a) |z| = 2 (b) |z + 2| = 3 Please specify the terms that you use if necessary and clearly explain each step of your solution.

### Residues and Poles : Cauchy Integral Formula

Find the value of the integral: {see attachment} taken counterclockwise around the circle (a) |z - 2| = 2 (b) |z| = 4 Please specify the terms that you use if necessary and clearly explain each step of your solution.

### Volume of Solid of Revolution (4 Problems)

52. Find the volume V of the solid with the given information regarding its cross-section: {see attachment for info and diagram) 55. Find the volume of the solid generated when the region {see attachment} on the interval {see attachment} is revolved about a) the x-axis b) the y-axis c) the line y = -2

### Volume of Solid of Revolution : Finding Volumes and Setting up Integrals (15 Problems)

1. Use shells to find the volume of the solid formed by revolving the given region about the y-axis: {see attachment for regions} 2. Set up but do not evaluate an integral using the shaded strips {see attachment} for the volume generated when the given region is revolved about a) the x-axis b) the y-axis c) the line y = -

### Find Areas and Sketch Bounded Regions (Curve)

2. Sketch a vertical or horizontal strip and find the area of the given regions bounded by specified curves: a), b), c) and d) {see attachment!} 3. Sketch the region bounded by and between the given curves and then find the area of each region: a), b), c), d), e) and f) {see attachment!}

### Lowest Common Multiples and Diophantine Equations

Please solve the following problems: 1. Compute the following ... 2. Let Fm be the set of all integral multiples of the integer m. Prove that ... 3. Draw the graphs of the straight lines defined by the following Diophantine equations ... 4. Prove that every integer is uniquely representable as the product of a non-negati

### Green's, Divergence and Stokes theorems Describe in 5-15 lines the links and connections among Green's theorem in all forms, Stokes' theorem and the Divergence theorem. In particular, your answer should address the question: Which theorem is an extension to which other theorem and in what way?

Real Analysis Divergence Theorem Green's theorem stokes' theorem

### Subring R of Integral Domain D is a Subdomain of D

Modern Algebra Ring Theory Subrings Integral Domain

### Green's Theorem and Stokes' Theorem

Using Green's Theorem and Stokes' Theorem respectively, calculate the given line integrals. • Using Green's Theorem calculate the line integral , where along the positively oriented closed curve C which is the boundary of the domain: . Which line integrals you would have to evaluate instead in order to calculate h

### Multiple Integration, Area, Center of Mass, Centroid and Jacobian

1.Given the region R bounded by y=2x+2 , 2y=x and 4. a) Set up a double integral for finding the area of R. b) Set up a double integral to find the volume of the solid above R but below the surface f(x,y) 2+4x. c) Setup a triple integral to find the volume of the solid above R but below the surface f(x,y)=-x^2 +4x. d) Set

### Find the work done by a force on a body along a curve.

Find the work done by a force F with F = (x,y,z) = (sinx, x+y, e^z) which results in the movement of a body along the curve C with parameterization r = (t, t^2, logt)for tE[1,2]. (See attachment for second question)

### Evaluate the path integral of a helix.

Let C be the helix, with parameterization r(t)=(cost, sint, t), tE[0,2pi] and let f(x,y,z) = x^2 + y^2 + z^2. Evaluate the path integral. (See attachment for full question) • Let C be the helix, with parameterization , and let . Evaluate the path integral . • Show that, if is a continuously differentiable conserv

### The problems are from complex variable class, 500 level in undergraduate.

The problems are from complex variable class. Please specify the terms that you use if necessary and clearly explain each step of your solution. If there is anything unclear in the problem, please tell me. Thank you very much.

### Cauchy-Goursat Theorem : Proving an Integral Equality

Please see the attached file for the fully formatted problems.

### Integral of a Principal Branch

Show that the integration from -1 to 1 z^i dz = ((1+e^-pi)/2)*(1-i)where zi denotes the principal branch... (See attachment for full question)

### Integration Problem Presented

∫ (from 0 to Infinity)(3(theta)^3*(x)^2)/((x+ theta)^4) dx

### Double Integral : Disc

By changing variables to polar coordinates evaluate the integral , where And , i.e., the disc of radius 3 centred at the origin. Please see the attached file for the fully formatted problems.

### Double Integral : Change of Variables to Polar Coordinate

Prove &#8747; 0 --->&#8734; e^(-x^2) dx = sqrt(pi)/2 Hint: multiply the integral with itself, use a different dummy variable y, say, for the second integral, write it as a double integral, and use change of variables to polar coordinate.

### Euclid's Division Lemma and Fundamental Theorem of Arithmetic

1. Without assuming Theorem 2-1, prove that for each pair of integers j and k (k > 0), there exists some integer q for which j ? qk is positive. 2. The principle of mathematical induction is equivalent to the following statement, called the least-integer principle: Every non-empty set of positive integers has a least element.

### Double Integral and Change of Order of Integration

I) Evaluate the integral.... ii) Change the order of integration and verify the answer is the same by evaluating the resulting integral. Please see the attached file for the fully formatted problems.

### Circle Desciribed in a Counterclockwise Direction : Find Limit of Integral Using L'Hopital's Rule

Please see the attached file for the fully formatted problems.

### Analysis of a Midpoint of a Line Segment

Let C denote the line segment from z = i to z= 1. By observing that, of all the points on that line segment, the midpoint is the closest to the origin, show that |&#8747;c dz/z^4| &#8804; 4 sqrt(2) without evaluating the integral. Please see the attached file for the fully formatted problems.