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Integrals

Indefinite Integral Square Root Functions

Find the indefinite integral (3-x)/sq root of 9-x^2 dx(dx would be in the numerator). I tried to split this problem apart. First part was: The integral of 3/sq root of 9-x^2 dx and found 3 arcsin x/3 + C, then Second part was: The integral of -x/sq root of 9-x^2 dx and found -3/4 -x + C. I then put them back together to ge

Analysis Proof Absolute Value

Note: If you have already answered this exact question please do not answer it again. I would like an answer from a different T.A. Thanks Say abs = absolute value. Suppose that the function f:[a,b]->R is Lipschitz; that is , there is a number c such that: abs(f(u) - f(v)) <= (c)abs(u-v) for all u and v in [a,b]. Let P

Picard's Method of Successive Approximations

Please see the attached file for the fully formatted problems. Attached is a file with a three part successive approximation problem. The following problems are to use the method of successive approximations (Picard's) [EQUATION] y x y fty tdt =+&#8747;n&#8722; with a choice of initial approximation other than y0(x)=y0

Infinitely Differentiable Function that is Not Analytic

Use the given information: the functions g:[a,b]->R and h:[a,b]->R are continuous with h(x) >= 0 for all x in [a,b], and there is a point c in (a,b) such that: the integral from a to b of h(x)g(x)dx = g(c) times the integral from a to b of h(x)dx. to show that the Cauchy Integral Remainder Theorem implies the Lagrang

Green's Theorem Evaluated Integral

Apply Green's Theorem to evaluate the integral over C of 2(x^2+y^2)dx + (x+y)^2 dy, where C is the boundary of the triangle with vertices (1,1), (2,2) and (1,3) oriented in the counterclockwise direction. Also check the result by direct integration. Please show detailed working so I can follow the steps of the working.

Indefinite Integrals for Anti-Derivative

Find the indefinite integrals (anti-derivatives): Find the indefinite integrals (anti-derivatives): a.) x / (x +2) dx I found ½ ln + + C as an answer - is this correct? b.) 1 / (x +2) dx I found 1/x arctan /x + C as an answer - is this correct? (I said that a = x, u = , du = dx )

Real Anaylsis

Let f: [a,b] be mapped onto the Reals be a function that is integrable over [a,b] and let g: [a,b] be mapped onto the Reals be a function that agrees with f except at finitely many points. Is g integrable over [a,b]? Why or why not?

Function integrable proof

Let f: [a,b] mapped onto Reals be a nonnegative function that is integrable over [a,b]. Then the integral from a to b of f = 0 if and only if greatest lower bound of f (I) = 0 for each open interval I in [a,b].

Sketching and Double Integration

Please see the attached file for the fully formatted problems. I.A. Sketch the following region in the x-y plane: R: 0<x<b^2 : x^1/2 < y< b B. Set up integral R for (e^-y2)/y dA in two ways.

Triple Integrals : Finding Volume of Solids with Boundaries

1) Evaluate the triple integral e^(1-(x^2)-(y^2)) dxdydz with T the solid enclosed by z=0 and z= 4-(x^2)-(y^2) 2) Find the volume of the solid bounded above and below by the cone (z^2) = (x^2) + (y^2), and the side by y=0 and y= square root(4-(x^2)-(z^2))

Integration: Cauchy-Schwarz Inequality

Suppose that the functions g:[a,b]-> R are continuous. Prove that: The integral from a to b of gf <= (the square root of the integral from a to b of g^2) multiplied by (the square root from a to b of f^2)

Quick Calculation of Laplace Integral

Please see the attached file for the fully formatted problem. Construct the quickest method to calculate the Laplace Integral. I = S e^(-x^2) dx infinity --> infinity

Evaluating Integral Functions

I'm taking a DE calculus class and I'm having problems figuring out the logic in solving some of the problems. The given integral is improper because both the interval of integration is unbounded and the integrand is unbounded near zero. Investigate its convergence by expressing it a sum of two intergrands-one from 0 to 1 an

Double Integration Transformation

Evaluate the double integral Transform the double integral of (i) using plane polar coordinates Show that the 3 x 3 determinant See attached file:

Integrals: Cost function and Marginal Cost

Given ist the following cost function: k(x)=x^3-9x^2+29x+35 x= quantity k= cost Question 1: At what quantity is the minimum of the marginal cost? Question 2: What is the increase of cost if the production is increased from 3 to 4 (integral)?

Evaluate integrals

Please see the attached file for the fully formatted problems. Evaluate the following integrals. S (4x^3 -2x - (2/x^3) dx S (1/2x^1/2) dx 1-->0 S ln x dx