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Integrals

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 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

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)

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.

Integrals : Riemann Sum

4. Let f(x) = 2x + 1 for 0 =< x =< 1. If the interval [0,1] is partitioned into 4 subintervals of equal length, then what is the smallest Riemann sum for f(x) and this partition? A. 7/4 B. 15/8 C. 2 D. 7/2 E. 7

Volume of a Hypersphere

Finding formulas for the volume enclosed by a hypersphere in n-dimensional space. c) Use a quadruple integral to find the hypervolume enclosed by the hypersphere x^2 + y^2 + z^2 + w^2 = r^2 in R^4. (Use only trigonometric substitution and the reduction formulas for &#8747;sin^n(x)*dx or &#8747;cos^n(x)*dx.)

Square Fan Blade Application of Double Integrals

Consider a square fan blade with sides of length 2 and the lower left corner placed at the origin. If the density of the blade is &#961;(x,y) = 1 + 0.1x, is it more difficult to rotate the blade about the x-axis or the y-axis? Please show steps.

U-Substitution, Integration by Parts and Differential Equations (8 Problems)

Please show all of the steps needed to solve the 8 integrals and differential equations that are attached. The integral of x(cos(x) dx The integral of (x^3) sin(x) dx The integral of t(csc(t))cot(t) dt The integral of arctan x dx The integral of e^2x sin(X) dx Solve the differential equation. y' = xe^x2 dy/dt = y

U-substituitions and integretion by parts

Please give step by step detailed solutions and answers to all. Please oh please do not skip steps. I need to see all these problems done so I can understand them and the way to think of solving them. I cannot find a one on one tutor and since I am auditing this course just for knowledge the teacher does not want to waste the

One Dimensional Riemann-Integrable

Q. Show that f is Riemann-integrable. What is &#8747;[0,1] f(x)dx? (Hint: What's the set of discontinuity of f? Does it have Vol1-zero?) Please see attached for full question.

Continuous Functions, Fundamental Set of Solutions

Consider the attached differential equation where I = (a,b) and p,q are continuous functions on I. (a) Prove that if y1 and y2 both have a maximum at the same point in I, then they can not be a fundamental set of solutions for the attached equation. (b) Let I = {see attachment}. Is {cos t, cos 2t} a fundamental set of solu

Calculating Traction from Stress Tensor Matrix

NOTE: in part A, the traction is just the integral of the dot product of T and n. 7 0 -2 The stress at point P = 0 5 0 -2 0 4 I want to know the traction vector on the plane at point P with the unit normal n = (2i1, -2i2, 1i3)/3

Centroid-triple integrals

Find the centroid of the first octant region that is interior to the to the two cylinders x^2+z^2=1 and Y^2+Z^2=1 centroid for x y and z are x'=1/M*triple integral of x^2*dV y'=1/M*triple integral of y^2*dV z'=1/M*triple integral of z^2*dV

Limits : Evaluating Integrals, Anti-Differentiation and Area Between Curves

(a) Consider the attached limit of summed terms (i) Explain why each of the sums in the attached expression gives an over-estimate of the area beneath the curve {see attachment} (ii) Evaluate this limiting sum, using the expression {see attachment} (iii) Check your answer in (ii), by using the fundamental theorem

Gravitational Attraction : Triple Integral

Find the gravitational attraction of a solid hemisphere of radius a and density 1 on a unit point mass placed at its pole REVIEW: Fz=G*triple integral of density*cos(phi)sin(phi)d(rho)d(phi)d(theta)

Changing variables in multiple integrals

Using the coordinate change u=xy, v=y/x, set up an iterated integral for the polar moment of inertia of the region bounded by the hyperbola xy=1 , the x-axis, and the two lines x=1 and x=2. Choose the order of integration which make the limits simplest THIS MESSAGE IS ADDRESSED TO ANY TA: I found something , I just want you

Control Systems Under Proportional-Integral Control

Consider a block diagram describing a system under proportional-integral control (as show in figure in attachment): Find the constraints and determine the range (using the Routh-Herwitz criterion) of Kp and Ki. Also, find the closed loop system transfer function assuming the controller gains are set to a specific value. (Ple

Function and Differential Equations

See attached explanation Differential equations are not my strong suit now. Please explain in a simple way each step from the integral 1/F dF to the final answer. Show and tell how you get from step to step. On problem 35 please answer and explain this in the simplest way you can for me to understand please. Step by

Limits of iterated integrals (parallel axis theorem)

Prim is primitive! In genral the moment of inertia around an axis( a line) L is: Isubl=double prim (dist(.,L)^2*delta*dA) The collection of lines parallel to the y axis have the form x=a .Let I=Isub(y) be the usual moment of inertia around the y axis I= double prim of x^2*delta*dA Let I(bar) be the moment of ine

Limits in iteratred integrals

Find the average area of an inscribed triangle in the unit circle.Assume that each vertex of the triangle is equally likely to be at any point of the unit circle and that the location of one vertex does not affect the likelyhood the location of another in any way. (note that the maximum area is achieved by the equilateral trian

Integration: The Limit of a Sum

Find by the method of summation the value of : a) The integral (from 0 to 1) of the square root of x. (dx) b) The integral (from 1 to 4) of 1 divided by the square root of x. (dx) Please view the attachment for proper formatting.