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Integrals

Indefinite integrals found

1. Find the indefinite integrals 2. Suppose that the rate of increase of paper production in the US for recent year is given approximately by... (see attachment)

Integral

Please evaluate the integral attached

Integration by parts

34.8 (a) Use integration by parts to evaluate 1 ∫ xּarctan x dx. 0 Hint: let u(x) = arctan x, so that u′(x) = 1/(1+x2). (b) If you used v(x) = x2/2 in part (a), do the computation again with v(x) = (x2+1)/2. This interesting example is taken from J. L. Borman[6].

Integral

Evaluate the integral from 0 to inf. of [cos(3x)-cos(5x)]/x^2 dx

One Dimensional Riemann-Integrable

Q. Show that f is Riemann-integrable. What is ∫[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.

Integral, Continuity and Limits

Please see the attached file for the fully formatted problems. Q: Suppose and are continuous and F(x) = Let (a) prove that f'(x) + g'(x) =0 for all x (b) Prove that f(x) + g(x) = /4 for all x. Deduce that

Integration : Fubini's Theorem

Use attached to solve the following question by integrating over an appropriate rectangle. Assume f is class C2 Prove the following theorem by Fubini's Theorem. Please see attachment. For f of class C2 Left Hand side: Right Hand side: Use above to solve the following question by integrating over an appropri

Integration : Class C2

Calculate to show, for f of class C2 ... {see attachment} What is the integral on the right equal to {see attachment}

Fubini Type I : Interpret Iterated Integrals as Triple Integral

Interpret the attached iterated integrals as a triple integral for the appropriate region {see attachment}, sketch {see attachment} and change the order of integration so that the innermost integral is taken with respect to y. (f is continuous) ... **See attachment for complete 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

Solve the ODE

Use words to describe the solution process. Typeset solutions. Work is to be done without the aid of a calculator or computer. Show all steps. For example, if you integrate by parts, show all the steps of integration. If you use an integral table, state that as well. Find the solution to y''' + 2y'' + y' = 0 satisfying

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

Joint Density

The joint denisty function of X and Y is given by [see attached] (a) Find E(X) (b) Find E(Y) (c) Show that Cov(X,Y)=1 *(Please see attachment for complete problem)

Area Between a Curve

Please assist me with the attached problems relating to finding the region within a curve. 3. (a) Obtain an expretsian far Calculating the area between the curve y=2?x+x2 and the u-axis far 0 <x< 2 by dividing the area up into 2n strips of equal width (each strip will have width 1/n) and then taking the limit as n ---> infini

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