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Integrals

The Riemann Stieltjes integral

See Attached If f is integrable on [a,b] with respect to f, then . a) Prove this by examining the two Riemann-Stieltjes sums for a partition obtained by taking . b) Prove this by using the Integration by Parts Theorem.

Volume of the solid obtained by rotating the region

1. For each of the following integrals do the following: (i) Explain in one sentence why the integral is an improper integral. (ii) Decide whether the integral is convergent or divergent. Justify your answer. Please see the attachment for the integrals. 2. Find the area of the region bounded by the curve y = 3x - x^2 a

How many gallons of paint will Amy need to paint both sides of the fence?

Amy plans to paint both sides of a fence whose base is in the xy-plane and is given by the points (x,y) where x = 30cos^3(t) and y = 30sin^3(t) for 0 <= t <= (PI/2), and whose height z at (x,y) is given by z = 1+(1/3)y, all measured in feet. How many gallons of paint will Amy need if a gallon covers 200 ft^2?

Riemann integrable function

1. Let f: [a,b] ----> R and suppose f is integrable with respect to alpha. Prove that for any c in the real numbers, cf is integrable with respect to alpha and the integral from a to b of cf d(alpha) = c times integral from a to b of f d(alpha). Give an example of a function f : [0,1] ----> R such that f is Riemann integra

line integration and Green's theorem

Compute the integral of 4y^3 dx -2x^2 dy around the square bounded by the lines y=+/- 1 and x=+/-1 a) directly by performing the line integration, and b) by using Green's theorem. By symmetry, it is obvious that one of the terms in the integrand of the above line integral can be ignored? Which term?

Indefinite Integral Typified

Evaluate: Integral Sign(top 4, bottom 0) dx _____ (2x + 1)^3/2 Evaluate: Integral Sign (top 4, bottom 1) (square root x + 1/square root x) dx Please explain every step.

Show that f_n(x) is Riemann integrable.

Let f_n(x)= (sin(1/x))^n when x does not equal zero and f_n(x)=0 when x = 0 . a. Show that f_n(x) is Riemann integrable. b. Calculate lim(as n goes to infinity) integral ( from 0 to 1) f_n(x)dx .

Inspecting the space of continuous functions

Let C[0,1] be the space of continuous real functions where C[0,1] has the following norm ||f||_2 =( integral (from 0 to 1) |f(t)|dt)^(1/2) Consider f_n(t) = 0 for t greater than or equal to zero and less than or equal to 1/2-1/n, f_n(t)=1+n(t-1/2) for t great than or equal to 1/2-1/n and t less than or equal to 1/2 a

Modelling the space of continuous functions

Let C[0,1] be the space of continuous real functions where C[0,1] has the following norm ||f||_2 =( integral (from 0 to 1) |f(t)|dt)^(1/2) Consider f_n(t) = 0 for t greater than or equal to zero and less than or equal to 1/2-1/n, f_n(t)=1+n(t-1/2) for t great than or equal to 1/2-1/n and t less than or equal to 1/2 a

Solving for a "Double Integral"

Question: Evaluate the double integral xy dA where R is the region bounded by the graphs of y= square root x, y= 1/2x, x=2, x=4

Polar Coordinates: Example Problem

Use polar coordinates to calculate the area of the first and second quadrants of the region that is bounded by the line y =- x and the graph of r = 3(1-cos??). Specifically, I want to know if the second integration intervals go from zero to pie?

double integral evaluated ..

Evaluate the double integral: a) SS (x^2 - 2y) dA R={(x,y) | 0<=x<=2 , 0<=y<=1} R b) S (x=0 to 1) S (y=0 to 2x) SS (x+2y) dydx S-denotes the integral symbol. On B, the first two S's are the limits.

Derivative of a distribution

Let f(x) and g(x) differentiable functions on R. Evaluate the derivative, in distribution sense, of the function h(x): h(x) = f(x) if x > 0 h(x) = g(x) if x â?¤ 0.

Convergence to Delta function

Please solve problem number 10 in part 1 ( Theorem ...) of the attached PDf file: "The sequence k^n e-pik^2IxI^2 converge to delta in distribution sense (See correct transcript of the problem as attachment)."

Integration of polynomial quotient and partial fractions

A example is shown whereby the velocity of an object is expressed as a quotient of two polynomials in time t. Such that velocity v = (t^2 + 18t + 21)/{(2t+1)*(t+4)^2} The problem asks to determine the displacement of the object after time t = 2.1s. The process requires the integration of the polynomial between the limit

Lebesgue integral

Let f be defined on the interval [0,1] by setting f(x)=0 if x is irrational, and if x=(m/n) rational where gcd(m,n)=1 set f(x)=n. Show that f is unbounded on every open interval in [0,1] and compute the Lebesgue integral of f over the interval [0,1].

Region Bounded by a Graph

Please help me find solutions to these problems. Please show your work along with explanations. Thank you for your help! 1. Find the area of the region bounded by the graph of , the -axis, and the line . 2. Find the derivative: . 13. Find

Finding the mass of a solid by integration

Set up different integrals to find the mass of the solid bounded by the equations z=8-2x, z=0, y=0, y=3 and x=0. the density of the solid at (x,y,z) is d(x,y,z)=kx, k>0. evaluate ONE of the integrals. sketch the solid and the projected regions in each of the xy-, xz- and yz- planes.

Integration and continuity

6.Prove that if g(x) is nonnegative and continuous on [0, 1] and (integral from 0 to 1 of g(x)dx) = zero then g(x)= 0 on [0,1] 7. If f is continuous on [0,1] and if (integral from zero to one of (f(x) x^n)dx)) =0 for n in the Naturals, prove that f(x)=0 on [0,1]/ Hint: The integral of the product of f with any polynomial is

Contour integral involving branch point singularity

By considering the integral of (z^2+1)^(-a) around a suitable contour C, prove: Integral from x=0 to x=infinity of dx/(x^2+1)^a = sin(pi*a) Integral from u=1 to u=infinity of du/(u^2-1)^a where 1/2 < a < 1. (Include proofs that the integrals over any large or small circular arcs tend to zero as their radii tend to in

The integral of each such simple function

Recall that the support of a function f : Rn â?'R is the closure of the set {x É? Rn : f(x) = 0}. Prove that if f : Rnâ?' Rn has support in a set of Lebesgue measure =0, then â?«Rn f(x)dx = 0. The original problem is written in PDF file sent as attachment below. It is number 3 on the list.

Volume of solids of revolution..

Volume of solids of revolution.. 1. A paraboloid dish (cross section ) is 8 units deep. It is filled with water up to a height of 4 units. How much water must be added to the dish to fill it completely? 4. Write an integral that represents the volume of the solid formed by rotating the region bounded by , , , and

indefinite integral calculus

1. A quantity of gas with an initial volume of 1 cubic foot and a pressure of 500 pounds per square foot expands to a volume of 2 cubic feet. Find the work done by the gas. (Assume that the pressure is inversely proportional to the volume; that is, the gas obeys the ideal gas law such that PV = constant.) 2. Find the ind

improper rational function

Integration by partial fractions 6. Solve for : 10. Evaluate: 11. Evaluate: 19. Express the improper rational function as the sum of a polynomial and a proper rational function:

Integration and Differentiation Exponential Functions

Please explain the solutions so that I can understand. 7. Evaluate y = Integration-sign e^(ax+b) dx . 11. Find the area of Recordi's new exponential pool, which is bounded by y=e^x, y=0, x=0 and x=3. 12. Find dy/dx for y=x^2 e^-x . Find points where the curve has horizontal tangents. 13. Find out how many bac

polar coordinates calculus

Use double integration in polar coordinates to find the volume of the solid that lies below the given surface and above the plane region R bounded by the given curve. 1. z=x^2+y^2; r=3 Evaluate the given integral by first converting to polar coordinates. 2. ∬_(0,x)^1,1▒〖x^2 dy dx〗 Solve by double i

Solid of Revolution Integration

Use the disk method to find the volume of the solid of revolution formed by revolving the region about the x-axis. Y = sqrt16-x Solid region is a semi circle from 0,4 to 16,0 find the solid amount between 5,0 and 6,0.

integration procedure

I need to find the center of mass (gravity) for problems two and tree. all integration procedure be shown . The answers required are. (the x and the y) 2a) x bar = 2b) y bar = 3a) x bar = 3b) y bar =