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Integrals

Inner product

Consider the functions below and the corresponding inner product are these functions orthogonal to each other with respect to the inner product? f(x)=cos x g(x)=(3 cos ^2 x-1) {INTEGRAL sign from 0 to pi} f(x) g(x) sin x dx

approximations to different integrals

Below are some approximations to different integrals. By extrapolating from these approximations give the value of the integral to a level of accuracy you feel is justified. • T2=4.02441, T4=4.09123. • M8=11.3421, M16=10.6745. • S1=4.4692, S2=4.9899.

Prove That Ind (Gamma) Is Always an integer

Let gamma be a continuously differentiable closed curve in the complex plane, with parameter interval [a,b], and assume that gamma (t) is not equal to 0 for every t in [a,b]. Define the index of gamma to be Ind (gamma) = 1/2pii integral from a to b of gamma prime (t) over gamma (t) dt. Prove that Ind (gamma) is always an integer

The Body Travels Only an Infinite Distance

Suppose that a body moves in a straight line through a resisting medium with resistance proportional to the square of it's speed v with initial speed v_0 and initial position x_0. Show that it's speed and position are given by v(t)=v_0/(1+v_0kt), x(t)=x_0+((ln(1+v_0kt))/k). Conclude that the body travels only an infinite distanc

Integration problems

I need help solving the following problems. Can you explain to me how to solve these problems? Find the derivative of y = log(2 + sin⁡〖x)〗 Find the derivative of f(x) = sinh2(3x) Find the derivative of y = x^(x-1) Find the derivative of y = ln⁡((1+e^x)/(1-e^x )) Evaluate ∫_0^(π/4)▒tan⁡

The Volume of the Solid by Rotating the Region around Y-Axis

Please explain in detail as much possible: Find the volume of a triangle with sides y = 2, y = x+1, y = 11-x. The vertices are (5, 6), (1, 2) and (9, 2). Find the volume by rotating about y-axis: (i) Method of disks (ii) Cylindrical shell method

a first order non homogenous equation

A tank contains 200 liters of fluid n which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 4L/min; the well-mixed solution is pumped up at the same rate. Find the number A(t) of grams of salt in the tank at time t.

Equilibrium solutions, Separation of variables, exact equations,

What technique would be used to solve the differential equation: You do not have to solve the differential equations, just write what technique that you would use to solve them. Solution techniques: Equilibrium solutions, Separation of variables, exact equations, integrating factors, substitutions for homogeous and Bernoulli

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.

Prove A has an Empty Interior

The set A = { f, such that f is an element of L^2 on [0,1] } can be considered as a subset of the metric space L^1 on [0,1]. Prove that A has empty interior as a subset of L^1 on [0,1].

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.