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Integrals

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 ∫sin^n(x)*dx or ∫cos^n(x)*dx.)

Definite Double Integral

Integrate the function (1/θ)^2 e^((-x1-x2)/ θ)) from 0 to 2 ln 2- x2 (for dx1) and from 0 to 2 ln 2 (for dx2).

Functions

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.