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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 =

Taylor polynomial, Lower and Upper sums

2a. Find the second order Taylor polynomial for f(x) = x^(1/3) about x = x0, x0 > 0. 2b. Show that the function G(x) is differentiable at x0 and find G'(x0). 3a. Find an expression for the lower sum L(f,D) and upper sum U(f,D). 3b. Determine the lower integral and upper integral. Please refer to the attached images f

line integral solving process

Please show the steps in solving process. Parts of the questions are given below to give a quick idea about them; please refer to the attachment for complete and all the questions. (1) Evaluate the line integral. (2) Let C be a circle of radius 1 centered at the point (x,y) = (1,0). Consider the vector field F(x,y) = xi +

Integration and Differentiation

1. Evaluate the integrals. a. Integral of cot cubed root of x / cubed root of x^2 dx b. Integral of cos^2x / sinx dx c. Integral of cosx sinx / cos^2x-1 2. Solve the differential equation subject to the given conditions. y'' = 6 e^2x ; y = -3 and y' = 2 if x = 0 3. Find f'(x) for each of the follo

Cauchy-Goursat Theorem

Given: Integral from zero to infinity of cos(x^2) dx = integral from zero to infinity of sin(x^2) dx = 1/2 * (square root of pi/2). These can be evaluated by considering cos(x^2) = Re(e^ix^2) and sin(x^2) = Im(e^ix^2). 1.) Integrate the function f(z) = e^i(z)^2 around the positively oriented boundary of the sector 0 <= r

Improper Integrals

1.) Use residues to evaluate the improper integral from 0 to infinity: 1 / (x^2 + 1)^2 dx 2.) Use Jordan's Inequality to evaluate the improper integral from -infinity to infinity: (x^3 sin ax) / (x^4 + 4) Thank you for your assistance.

Calculating Integrals

Evaluate the integral. *** Integral is attached *** Note: Integral is followed by dx This *should be* "integral of 4x^2 times the fourth root of (8 + 4x^3) times dx" and each of the smaller numbers in the choices below are the exponents. Second Attachment should be here! a. -8/3(8+4x^3)^-3/4 + C b. 4(8+4x^3)^5/4

Fundamental Theorem of Line Integrals

a). Show that the line integral ∫_C▒〖ysinxdx-cosxdy〗 is independent of the path. b). evaluate the integral in part (a) along the line segment from (0, 1) to (π,-1) c). Evaluate the integral ∫_((0,1))^((π,-1))▒〖ysinxdx-cosxdy〗 using Theorem 16.3.1 and confirm the value is the same as that obtained in part

Stable Steady-State Value

Solve the following linear, first-order differential equations and ensure that the initial conditions are satisfied. Show whether or not the steady-state solutions are stable. (a) 10y' = 5y and y(0) = 1. The answer is y(t) = e^(1/2t), but having trouble arriving at that answer (b) 4y' - 4y = -8 and y(0) = 10. The answer

Integration of trigonometric functions

I am supposed to find or evaluate the integral of cos^(2)XsinXdx I was thinking that i should use integration by parts but I am not sure what values to use for U, and dv...

contradiction to the condition

Let (X, E, u) be a measure space with u a non-negative measure. Suppose that 1. f : X -> R is measurable 2. f (x) >= 0 a.e. with respect to u. 3. integral (over X) f du = 0 Prove that f (x) = 0 a.e. with respect to u. note: E = Capital Sigma u = lowercase mu

Trapezoidal Rule Word Problems

Please show ALL work! A bacteria population grows at a rate proportional to its size. Initially the population is 10,000 and after 5 days it's 30,000. What is the population after 10 days? How long will it take for the population to double? A solid S is generated by revolving the finite region bounded by the y-axis

derivative and the slope of a curve

In this explanation, I am being asked to discuss the relationship between the slope of a secant line, the slope of a tangent line and the derivative AND in addition, I must explain the relationship between the area of a finite number of rectangles under a curve and an infinite number of rectangles under a curve and the definite

Differential Equations and Indefinite Integrals

1. Find the solution of the differential equation dy/dx=x^3 -x, with the initial condition y(0) = -3. 2. Find the solution of the differential equation dy/dt=t^2 / (3y^2), with the initial condition y(0) = 8. 3. Find the solution of the differential equation dy/dx=(x=2)y^(1/2), with the initial condition y(0) = 1. 4. Fi

Calculaion of work done on a spring

1) Find the arc length of the curve f(x)=x3/2 - 1 over the interval [0,1] 2) Find the arc length of the curve f(x)=ln(cos x) over the interval [0,pi/4] 3) Find the arc length of the curve f(x)=1/6 x3 + 1/2 x - 1 over the interval [1,2] The Next TWO Problems Refer To The Paragraph Below. This is a two step module. First you

Defining a Function: Example Problem

A rational number r = p/q, where p, q are in Z, is said to be properly reduced if p and q (q > 0) have no common integral factor other than +1 or -1. Define the function f as follows; f(x) = q , if x = p/q, properly reduced. f(x) = 0 , if x is irrational. Prove that for every real number x, f fails to be bounded at x.