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# Integrals

### Evaluating an integral in terms of areas

"Evaluate the integral by interpreting it in terms of areas: the integral as 0 goes to 8 of |5x-10|dx"

### Use Definition of Integrals to Evaluate an Integral

Use the definition of integrals to evaluate the following integral: the integral as 1 goes to 8 of (2+3x-x^2)dx

### Properties of Integrals : Verify an Inequality

Use the properties of integrals to verify the inequality without evaluating the integral: [the integral as 1 goes to 2 of (sqrt(5-x))dx] is greater than or equal to [the integral as 1 goes to 2 of (sqrt(x+1))dx].

### Integrals : Riemann Sum with Diagrams

This question has me going around in circles. I can't make the Sigma symbol on the computer, so I used the word "Sigma" instead. For (c), n is above the Sigma symbol and i=1 is below it. (a)Find an approximation to the integral as 0 goes to 4 of (x^2-3x)dx using a Riemann sum with right endpoints and n=8. (b)Draw a diagram

### Elementary Numerical Analysis

GAUSSIAN NUMERICAL INTEGRATION 1. Consider approximating integrals of the form... in which f(x) has several continuous derivatives on [0, 1] a. Find a formula... which is exact if f(x) is any linear polynomial. b. To find a formula... which is exact for all polynomial of degree &#8804; 3, set up a system of four e

### Evaluate the integral.

&#8747;x/[x + (x^1/2) -2]

### Improper Integrals; Hyperbolic Functions; Convergence and Divergence etc.

Please assist me with the attached problems, including: Show that the improper integral converges and find its value or show that it diverges ... Please see attachment for complete list of questions. Thanks

### Integrals; Exact Value; Volume; Length (30 Problems)

Please assist me with the 30 attached problems, including: - Finding integral - Finding exact value - Finding volume - Finding length

### Integrals; Sine; Cosine; Bounded Region etc.

Please assist me with the attached problems. Examples: 2. Find each integral 5. Integrate equations using tables 6. Derive the sine squared formula 42. Use substituion to integrate certain powers of sine and cosine 52. Find the area of the region bounded by the graphs etc. (see attachment)

### Triple Integral : Find the mass of a prism given a density function.

Find the mass of the rectangular prism .... with density function ... where m = triple integral of density. Please see the attached file for the fully formatted problem.

### 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. &#8747; (2x-3)(x+2)dx 2 2 2. &#8747; (2x+1)4dx 1 0.001 3. &#8747; 50cos50&#960;t dt -0.001 &#960;/2 4. &#8747; 5sin(2t-(&#960;/6)dt 0

### Residues and Poles : Cauchy Integral Formula (Counterclockwise around a Circle)

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.

### Find Areas and Sketch Bounded Regions (Curve) (10 Problems)

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!}

### Fundamental Theorem of Arithemtic : 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

### Abstract Algebra/ subdomain 1) If D is an integral domain and R< = D is a subring of D with unity, show that R is a subdomain of D. This amounts to showing that 1subR and 1subD are unities from R and D respectively 2) Give an example with D not integral domain where 1subR =/= 1 sub D (Hint: consider Rsub1 X Rsub2. Warning Rsub1 is not a subring of Rsub1 X Rsub2; it's not even a subset of it) Note : R <= D means R is a subring of D 1subR means 1 is element of R 1subD means 1 is element of D Rsub1 mean that R subscrip1 Rsub1 X Rsub2 means cross product.

Modern Algebra Ring Theory Subrings Integral Domain

### Multiple Intergration, Area, Center of Mass, Moment, 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)

### 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

&#8747; (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.

### 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.

### Integral of a Semicircle and Segment

F(z) = z - 1 and C is the arc from z = 0 to z = 2 consisting of (a) the semicircle z = 1 - e^(i&#952;) (pi &#8804; &#952; &#8804; 2pi) (b) the segment 0 &#8804; x &#8804; 2 of the real axis. Find the integral &#8747;c f(z) dz for the two cases.

### Double Integral : Area of a Triangle

Please see the attached file for full problem description.