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    Integrals

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    Integration to Determine Given Area

    For 21 and 22 on the attached page, please set up a definite integral that gives the area of the region. (Do not evaluate the integral). Please offer as much explanation as possible.

    Volume and integration

    Find the region enclosed by x=3y and x=-y^2+4. Set up integrals both shell and disc that represent the volume generated when this region is revolved about y=4. Set it up, do not work to completion.

    Integration using the Midpoint Rule

    Please explain and solve the following. Use the midpoint rule with n = 4 to approximate the area of the region bounded by the graph of the function and the x axis over the indicated interval. f(x) = x^2 + 3 [0,2]

    Integration

    Please explain how to solve and solve the following problem. Use the limit process to find the area of the region between the graph of the function and the y-axis over the indicated y interval. Sketch the region. f(y) = 3y, 0 is less than or equal to y which is less than or equal to 2

    Integration

    Please explain how to solve the following problems and solve the problems. Use the limit process to find the area of the region between the graph of the function and the x-axis over the indicated interval. Sketch the region. 47. y = -2x + 3 [0,1] 49. y = x^2 + 2 [0,1] 53. y = 64 - x^3 [1,4] 55 y

    Integration Sum of Terms

    Please explain how to solve the attached problems (as much explanation as possible) and solve to the specified answers. Find a formula for the sum of n terms. Use the formula to find the limit as n approaches infinity.

    Integration of Unit Squares Explanation

    Please explain how to do problem 23 on the attached scan. Answer is that the shaded region falls between 12.5 square units and 16.5 square units. As much explanation as possible please. Please also explain how to do 27, 28, 29 and 30 on the attached scan. As much explanation as possible. Please solve the problems. Answer

    Lebesgue Measures, Limits and Integrals

    Prove or disprove the following: If f is in L^1[0,1], then limit the integral over [0,1] of x^n*f = 0 as n goes to infinity. I saw a similar example asking to prove that the integral from 0 to 1 of x^2n f(x) dx = 0, and they used algebra of functions generated by {1,x^2}, but we haven't talked about that, so please when you

    Lebesgue Measures, Integrals and Limits

    Let f_n(x) = n^1/2 * x * e^(-n*x^3), for n = 1,2,3... (i) Find the maximum value assumed by f_n in the interval [0,1]. (ii) Find Lim (n -> infinity) of integral from 0 to 1 of (f_n(x))dx. All integrals here are with respect to Lebesgue measure. Please justify every step and claim. e here is the exponential function.

    Lebesgue Measures and Integrals

    Let {f_n} be a sequence of nonnegative Lebesgue measurable functions on [0,1]. Suppose that: (i) f_n -> f in [0,1] and (ii) integral over [0,1] of f_n =< K for all n and some constant K. Then f is in L^1[0,1] and || f||_1 =< K. All integrals are with respect to Lebesgue measure.

    Lebesgue Measures and Integrals

    Let a,b be real numbers such that 0 < a < b < infinity. Does the limit lim of ( integral from a to b of n*sin (x^2/n) dx , n is positive integer exist? ( prove or disprove). Find the limit if it exists. Prove all assertions and justify every step. The integral here is with respect of Lebesgue measure.

    Integration, proofs, spline interpolation

    7. This problem generalizes the factorial function, as in n!=n(n-1)(n-2)...(2)(1), to more general arguments than just the positive integers. (a) Use integration by parts to show that for any positive integer n, the integral with respect to x from 0 to infinity of xne-x is n! (b) Make a clear case that the integral exists

    Reimann Sum/ Integrals

    Find exact values for Riemann sums approximating the integral of the function f(x)=x2 on the interval [0,1]. Split up the interval into N equal segments, and find the upper sum (taking the maximum function value in each segment) and the lower sum (taking the minimum function value in each segment). You will need a mysterious loo

    Volume of cross vault

    Let two long circular cylinders, of diameter D, intersect in such a way that their symmetry axes meet perpendicularly. Let each of these axes be horizontal, and consider the "room" above the plane that contains these axes, common to both cylinders. (In architecture this room is called a "cross vault".) The floor of the cross vau

    Approximate Integrals using the Trapezoidal and Simpson's Rule

    1. Approximate the integrals using the Trapezoidal rule. a) Integral from -0.5 to 0 x ln(x+1) dx b) Integral from 0.75 to 1.3 ((sin x)2 - 2x sin x +1) dx 2. Find a bound for error in question 1. using the error formula, and compare this to the actual error. 3. Repeat question 1. using Simpson's rule 4. Repeat ques

    Complex integration using series expansion of analytic functions

    I want to check my answer: Evaluate the following integrals: integral over gamma for (sin z)/z dz, given that gamma(t) = e^(it) , 0=<t=<2pi ( e here is the exponential function) My work: sin z = z - z^3/3! + z^5/5! + ... + (-1)^n (z^(2n-1))/(2n-1)! + ... divide by z we get (sin z)/z = 1 - z^2/3! + z^4/5!

    Complex Integration using power series expansion of analytic functions

    Evaluate the following integrals: (a) integral over gamma of (e^z - e^-z)/(z^n) dz, where n is positive integer and gamma(t) = e^(it), 0 =< t =< 2 pi (b) integral over gamma of (dz/(z^2 + 1) ) where gamma(t) = 2e^(it), 0 =< t =< 2pi ( Hint: expand (z^2 + 1)^-1 by means of partial fractions PLEASE USE POWER SERIE

    Triple Integral on Specific Solid

    Compute a triple integral over a specific solid that would be very difficult in rectangular coordinates, but easy in parabolic cylindrical coordinates, u, v, z, where x=(1/2)(u^2-v^2) y=uv z=z You must come up with the solid. Remember that the Jacobian determinant (u^2+v^2) must be used when transforming an integral to this