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Find volume of the solid of revolution formed when the region bounded by y=sinx and the x axis for 0 < or=z< or =pi is revolved about the x axis. use disc method and shell method for this problem.


Find: 1 ∫(x^2+1)e^-x dx 0 Find: ∫2x.tan^3(x^2).(secx^2)^-1/2dx

Mathematics - Calculus - Integration in Commerce

Profit over the useful life of a machine - Suppose that when it is t years old, a particular industrial machine generates revenue at the rate R'(t) = 6,025 - 8t^2 dollars per year and that operating and servicing costs accumulate at the rate C'(t) = 4,681 + 13t^2 dollars per year. a) How many years pass before the pro

Differential equation

See attached for formatting 1. Considering the differential equation y' = (y/x)3 : a. Discuss existence and uniqueness of solutions. b. Determine if exist constant solutions. c. Determine the general integral. d. Solve Cauchy problems y(3) = -1, y(3) = 0 and determine the maximal interval of solutions. 2) Integrate the


A group has just been formed with an initial membership of P0. Suppose that the fraction of the membership of the group that remain in the group for at least t years is given by the function S(t), and that at time t, new members are being added to the group at the rate of R(t) members per year. Show that N years from now, the gr

Integral equation

Please help working on these problems Please show all steps section 7.7 # 18 See attached Solve the given integral equation or integro-differential equation.

Indefinite Integral

Compute the following definite integral: ___ ____ Integration limits: 1<x<2 Function to integrate: exp[sqrt(x+1)] / sqrt(x+1) see file for better representation.

Partial fraction integral

See attachment Compute the following definite integral using partial fraction decomposition: &#643;&#755;&#737; 5x - 2 / (x^2 - 8x + 12) ( &#61663; that's a 1 for upper limit and 0 for lower limit)

Integration by parts

See attachment Use integration by parts to compute the following definite integral: &#643; (0 is the lower limit of integration & 1 is the upper limit of integration) t ln (2t + 1) dt Hint: Make sure that you keep in mind that t is the independent variable

Initial value an derivation of parameters

Please show all steps to solution. Solve the initial value problem Where in three different ways. 1) Using the method of variation of parameters .You can use any standard table of integrals to perform any of the integrals required in the solution.

Integrable functions

Assume that fn->f uniformly on [a,b] and that each fn is integrable. Then f is integrable and lim n->inf of the integral from a to b of fn = integral from a to b of f

Evaluating Integrals with Green's Theorem

Use Green's theorem to evaluate the line integral along the given curve. I will be using the S sign to portray the integral sign. 1) S_c F*dr where F(x,y) = (y^2 - x^2 y)i + xy^2 j and C consists of the circle x^2 + y^2 = 4 from (2, 0) to (sqrt(2), sqrt(2)) and line segment from ( sqrt(2), sqrt(2) ) to (0, 0) and from (0,


1. a. Is R= {a+b(squareroot of 2): a,b element of Z} a domain? b. Using the fact that alpha= (1/2)(1 + (square root of -19)) is a root of ((x^2)- x + 5), prove that R={a + b(alpha) : a,b element of Z} is a domain. Z= integers 2. Assume that (x-a) divides f(x) in R[x]. Prove that (x-a)^2 divides f(x) if and only if

Evaluate integral

Evaluate integral C (y^2 + 1) dx + 2xy dy; (a) C is the straight line from (-1; 0) to (1; 0). (b) C is the arc of the circle x^2 + y^2 = 1 going counter-clockwise from (-1; 0) to (1; 0).

Volumes by Cylindrical Shells and Volume by Disks and Washers

First, I can't tell the difference between the two. Both seem to indicate a region bounded by two or three certain functions or points. I was wondering if anyone could give a step by step explanation of how to do 1. Volumes by Cylindrical Shells 2. Volumes by Disks and Washers

Numerical Analusis Problem using Matlab

Write a function to calculate the integral from a to b of f(x) dx using the composite trapezoid rule with n equal subunits. test the code on a) the integral from 0 to pi of sin(x), b) the integral from 0 to 1 of exp(x) and c) the integral from 0 to 1 of arctan(x). Provide the codes used and all the results and work.

Complex Integration

I would appreciate any help on how to do this, thanks! Please see file attached... &#61543;(z,r) is a circular contour of radius r>0 centred at z Explain why cannot be evaluated by applying Cauchy's integral formula with , when = 1. Hence evaluate the integral.

Evaluating indefinite and definite integrals

See attached file for full problem description. Please show in detail how to solve the following integral. (a) Please show in detail how to solve the following indefinite integral. (b) Thank you.

Complex Integration

Help required on how to do the following: to evaluate : integral (z+i)/(z+2i) dz where curve is semicircle from 0 to 4i, initial point 0 and where curve is circle |z-2i|=2 anticlockwise.

Complex integration

Please show all steps to solution. Let . Evaluate the integral dz Where &#947; is the unit circle.

Determining area under 2 curves

Please see attached file. I need to determine the area under each of the curves in red --- I have fitted a curve using Excel to derive the polynomial equation. In the first graph, the x-axis limits are 0 and 4.5; in the 2nd curve, the x-limits are 0 and 5.0. The units for the x-axis is time (minutes) and the y-axis is p

Line integrals

See attached page Evaluate the line integrals, where C is the given curve


See attached page Evaluate the integral by making an appropriate change of variables.