### Characteristic, inductive proof

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I need help on how to work out the solution to a function using the 'D-contour' (see attached file).

Find the integral of f(x) = x tan^-1 x/(1+x^2)^2

(See attached file for full problem description with proper symbols) --- Let and for (a) Use integration by parts to show that in for . Deduce that for (b) Compute for and verify that ---

1).If A is a subset of B, A,B in m ( measurable sets) then show that integral (region A) s dM =< integral ( region B) s dM Where s here is a simple non-negative measurable function. ( Please don't confuse this with bounded measurable functions, I need the proof for SIMPLE functions). 2). If E are measurable, X_E is the c

Please see the attached file for the fully formatted problems.

36)After t weeks, contributions in response to a local fund raising campaign were coming in at a rate of 2000te^-0.2t dollars/week. How much money was raised during the first five weeks. 38) Find the volume of the solid generated when the region under the curve y=sinx+cosx on the interval [0,pi/4] is revolved about the y axi

Find the area of each polar region enclosed by f(theta) for a <=theta<=b 36) f(theta) = theta/pi, 0<=theta<=2pi PLEASE SHOW EVERY STEP IN SOLVING THESE-NO COMPUTER PROGRAMS PLEASE. 4) Identify each curve as cardiode, rose(state # of petals), leminscate, limacon, circle, line or none of above. a) r=2sin2theta b) r^2=2c

Sketch the region bounded between the given curves and then find the area of each region for 16 and 22. 16) y=x^2+3x-5, y=-x^2+x+7 22) x axis, y=x^3-2x^2 -x+2 28) Find the area of the region that contains the origin and is bounded by the lines 2y=11-x and y=7x+13 and the curve y=x^2-5. Please see the attached file f

4. The demand curve for a product has equation p=20 e^(-0.002q) and the supply curve has equation p=0.02q + 1, where q is the quantity and p is the price in $/unit. a) Which is higher the price at which 300 units are supplied or the price at which 300 units are demanded? Find both prices. b) Sketch the supply and deman

6. The demand curve for a product has equation p = 100 e^(-0.008q) and the supply curve has equation p = (4√q) + 10 , where q is the quantity and p is the price in dollars/unit. a) At a price of $50, what quantity are consumers willing to buy and what quantity are producers willing to supply? Will the market push price

A service station orders a 100 cases of motor oil every 6 months. The number of cases of oil remaining t months after the order arrives is modeled by f(t) = 100e^(-0.5t) a) How many cases are there at the start of the six-month period? How many cases are left after the end of the six-month period? b) Find the average number

Evalutate the following: 1.) Integrate sech ^2x/ (2+ tanh x) dx 2.) Integrate from 0 to (Pi/2) sinx/(1+cos^2 x) dx 3.) Find (f^-1)' (a) of f(x)=x^5 - x^3+ 2x, a=2 4.) Find the limit x approaches (2-) e^(3/(2-x))

Use the convolution integral method and hand calculation to come up with the exact formula for the solution of y'' [t] + 5y' [t] +6y[t]= 3.8E^(-t) with y [0]=2 y' [0]= -1

#22) Find the moment of inertia of a lamina S of density 1 about an axis A, where S: x2+ y2=1, A: the line z= h/2 in the xz-plane (See attached file for full problem description with equations) --- Question in Kreyszig's Advanced engineering mathmatics 8th ed.: section 9.6: Surface integrals

#16) Surface integrals; s G(r) dA. Evaluate these integrals for the given data. (show the details.) G=(x2+ y2)2 - z2, S: r=[u cos v, u sin v, 2u], 0 u 1, -  v   (See attached file for full problem description with equations) --- Kreyszig's Advanced e

#12) Surface integrals; s G(r) dA. Evaluate these integrals for the given data. (show the details.) G=cosx + siny, S: the portion of x+y+z=1 in the first octant (See attached file for full problem description with equations) Question in Kreyszig's Advanced engineering mathmatics 8th ed.: section 9.

(See attached file for full problem description with equations and diagram) --- Compute ∫r+ (bar-z + z^2 bar-z) dz where gamma+ is a square with side = 4, centered at the origin and traced counterclockwise once ---

Let f(x) be a positive continuous function on [0,1/2], f(x) =< 1/2. Let A = { (x,y) : 0 =< x = 1/2, 0=<y=< f(x)} Prove that; m*(A) = integral from 0 to 1/2 of f(x)dx. Please I don't want a solution from a book, I want a simple proof based on basic definitionsand given info.

Find the volume of y = 1/sqrt(1+x^2) bounded by y=0, x=-1, x=1 I'm using the disc method with a dx function: V = pi integral( [R(x)]^2 ) dx Therefore, I have V = pi integral( [1/sqrt(1+x^2)] ^2 ) dx from -1 to 1 = pi integral( [ 1/(1+x^2) ] ) dx from -1 to 1 I can't figure out how to integrate. Please explai

I need to find the arc length of y = 1/6 x^3 + 1/(2x) on the interval [1,3]

Find the arc length of the graph of the function over the indicated interval: y=1/6 x^3 + 1/(2x^2), [1,3] I know S = Intergral( sqr( 1 + [f'(x)]^2 )) dx from 1 to 3. I get y' = [ 1/9 x^4 - 1/3 + 1/(4x^4) ] dx Therefore, S = Intergal( sqr( 1 + 1/9 x^4 - 1/3 + 1/(4x^4) )) dx from 1 to 3 = Intergal( sqr( 2/3 + 1

Let be a positive function in . Define a new function by Prove that . Please see the attached file for the fully formatted problems.

Find the integral of: ((1+(sinh t)^2)^(1/2))dt.

Let X be a normed space, I closed interval ( or half-open on the right) and a = inf I, b = sup I. Let h : I -> [0,infinity) be a continuous function such that integral ( from a to b ) h(t)dt < positive infinity where integral from a to b represents the improper integral when I is not closed. Let epsilon > 0 and

If is a measure space and , show that defines a bounded integral operator. Please see the attached file for the fully formatted problems.

Please see the attached file for the fully formatted problems.

Find the volume of the solid formed when the region described is revolved about the x axis using washers and disks. 14) the region under the curve y= cubed root of x on the interval 0≤x≤8. 16) the region bounded by the lines x=0, x=1, y=x+1, and y=x+2. 20) the region bounded by the curves y=e^x and y=e^-x on

Which is the acceptable trial solution? (See attached file for full problem description)

Show that the solution of: dM/dt = Poert - pM M(0)=0 is M(t) = [Po/(r + p)](ert - e-pt) Please see the attached file for the fully formatted problems.