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Which of the following conditions are necessary for a function f to be Riemann integrable on the closed interval [a,b], where a < b? I. f is bounded on [a,b]. II. f is continuous on [a,b]. III. f is differentiable on [a,b].

Maximum Value of a Function

Please see the attached file for the fully formatted problems. 2. If f(x) =&#8747; (sin t)^(1/3) dt pi/2 --> x then at what value of x in the interval [0, 2pi] is f(x) a maximum?

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

Real Analysis : Using a Summation Series to Estimate an Integral

Say the only tool you have available to you is a pocket calculator which performs addition, subtraction, multiplication, and division, accurate to 15 decimal places. Explain a practical way to compute: Integral from 0 to 1 of e^[-(x^2)] to within an error less than 10^-8. Prove that the method works.

Indefinite integrals

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)

Integration by parts

34.8 (a) Use integration by parts to evaluate 1 &#8747; x&#1468;arctan x dx. 0 Hint: let u(x) = arctan x, so that u&#8242;(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].

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.

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


See attached... Let g(a) be the real solution to x+x^5=a

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