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# Integrals

### Question Regarding Polynomials

(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.

### Measurable Spaces and Properties of Integrals of Simple Functions

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

### Integration: Profit Function, Pesent Values and Future Values

An oil company discovered an oil reserve of 100 million barrels. For time t>0, in years, the company's extraction plan is a linear declining function of time as follows: q(t)=a-bt Where q(t) is the rate of extraction of oil in millions of barrels per year a time t and b= 0.1 and a =10 . a) How long does it take to exhaust

### 12 Problems - Integrals : Sum of Partial Fractions and Volume of Solid of Revolution

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

### Area Bounded by Curves & Volume of Solid of Revolution

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

### Using Integrals to Find the Area Bounded Between Curves

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

### Integrals Application Word problem : Supply and Demand Curves and Equilibrium

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

### Definite Integrals : Supply and Demand Curves and Equilibrium

6. The demand curve for a product has equation p = 100 e^(-0.008q) and the supply curve has equation p = (4&#8730;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

### Calculate integral using Gauss theorem

Calculate integral using Gauss theorem. See attached file for full problem description.

### Definite Integrals Application Word Problem : f(t) = 100e^(-0.5t)

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

### Integrals Trigonometric Functions

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))

### Convolution integral method

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

### Graphing and TI-83 : Plotting a Graph an Calculating an Integral

I need help with the TI-83. I dont want the problem solved, just need to be walked through the steps for inputting data into the TI-83 to mimic the graph as shown in the document. I have tried many ways and I keep getting the wrong graph. Thank You. a) Graph x3 - 5x2 + 4x , marking x = 1,2,3,4,5 b) Use th

### Integrals, Marginal Cost Function and Marginal Profit

The marginal cost function of producing q mountain bikes is a) If the fixed cost in producing the bicycles is \$2000, find the total cost to produce 30 bicycles b) If the bikes are sold for \$200 each, what is the profit (or loss) on the first 30 bicycles c) Find the marginal profit on the 31st bicycle. Please see the atta

### Graphing and interpretation of definite integrals.

A) Graph , marking x = 1,2,3,4,5 b) Use the graph and the area of interpretation of the definite integral to decide which of the five numbers for n = 1,2,3,4,5 is largest. Which is smallest? How many of the numbers are positive? (Do not calculate integrals). (See attached file for full problem description)

### Vector calculus: Surface Integral, Moment of Inertia of a Lamina

#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

### Vector calculus: Surface Integrals

#16) Surface integrals; &#61682;s&#61682; 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&#61603; u &#61603;1, -&#61520; &#61603; v &#61603; &#61520; (See attached file for full problem description with equations) --- Kreyszig's Advanced e

### Vector Calculus: Surface Integral

#12) Surface integrals; &#61682;s&#61682; 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.

### Line Integral and Complex Form of Green's Theorem

(See attached file for full problem description with equations and diagram) --- Compute &#8747;r+ (bar-z + z^2 bar-z) dz where gamma+ is a square with side = 4, centered at the origin and traced counterclockwise once ---

### Continuity and Outer Measure

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.

### Integration : Finding Volume of a Disc

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

### Integration : Finding the Arc Length on an Interval

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

### Integration: Find the Arc Length Over an Interval

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

### Harmonic Analysis, Convolution and L^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.

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

### Normed Space, Compactness and Transformation

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

### Measure Space and Bounded Integral Operator

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

### Integrals (4 Problems)

Please see the attached file for the fully formatted problems.

### Volumes of solids

Use shells to find the volume of the solid formed by revolving the given region about the y-axis. 22) the region bounded by the curve y=SQRT(x), the y-axis, and the line y=1. 24) the region bounded by the parabolas y=x^2, y=1-x^2, and y axis for x&#8805;0. 26) the region inside the ellipse 2(x-3)^2 + 3(y-2)^2 = 6 about

### Volumes of Solids: Washers and Disks

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&#8804;x&#8804;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