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Integrals

PID (Proportional / Integral / Derivative) Control System

Consider the system shown in Fig.1. (attached file) This is a PID control of a second-order plant G(s). Assume that disturbances D(s) enter the system as shown in the diagram. It is assumed that the reference input R(s) is normally held constant, and the response characteristics to disturbances are a very important consideration

Integrate

Use partial decomposition to integrate (x-7)/(x^2-x-12 ) (5x)/(2x^3+6x^2 ) (x^3-8x^2-1)/(x+3)(x^2-4x+5)

Integrate

Use partial decomposition to integrate. (2x^2-x-20)/(x^2+x-6 ) (sint)(4cos^2t-1)/(cost)(1+2cos^2t+cos^4t) (x+pi )/(x^2-3pix+2pi^2)

Ski Jump Problem (projectile)

At time t = 0 a skier leaves the end of a ski jump with a speed of Vo feet per second at angle alpha with the horizontal. Position vector of the skier is represented by the vector : r(t) = [Vo (cos theta) t] i + [h + (Vo sin theta) t - 1/2gt^2] j The skier lands 259 feet down the incline 2.9 seconds later. (g = 32 ft. per sec^2

Integration Problems

-2 1. Evaluate ∫ (e3x - 4 / ex) dx -3 2. The mean value of a continuous function over a given range is defined as the integral of a function divided by the range. b

Integral Equation

Solve the integral equation: y(x) = 1 + integral from 0 to x of ( y(t)*t^2 dt) (in the integral is y as a function of t times t^2)

Convolution Sum : Viciously Damped Single-degree-of-freedom System

Derive the response of a viciously damped single-degree-of-freedom system of force Ft=F0e^-αt u(t) by means of a convolution integral. Plot the the response for the system parameters m=12 kg, c= 24 N.s/m, k=4.800 N/m and the force paramters F0=200 N, α=1 Prescribed Textbook: Fundamentals of vibrations : Leonard

Convolution Integral : Viciously Damped Single-degree-of-freedom System

Derive the response of a viciously damped single-degree-of-freedom system of force Ft=F0e^-αt u(t) by means of a convolution integral. Plot the the response for the system parameters m=12 kg, c= 24 N.s/m, k=4.800 N/m and the force paramters F0=200 N, α=1 Prescribed Textbook: Fundamentals of vibrations : Leonard

Heat transfer and Heat Equations

3-3. Use the methods of vector calculus to derive the general heat conduction equation (Hint: Apply the first law to a volume V with surface S. and use the Gauss divergence theorem to Convert the surface integral of heat flow across S to a volume integral over V.) The cylindrical and spherical coordinate systems are examples o

Integration using trigonometric substitutions

Please evaluate the following integral using trigonometric substitution. Please show each step in the solution. Then evaluate the answer for the limits 0 less than or equal to t less than or equal to ln4 ________ Integral e^t dt/ _/(e^2t +9)

Convergence of Integrals

Hello. I'm trying to show if the triple integral: (z^2)/[(x^2+y^2+z^2)^(3/2)] dV converges or diverges over the region: x^2 + y^2 + z^2 <= 1. And if it converges, to what value? I am curious as to how I would do this, and how does one generally show if an integral converges or diverges? Thanks

Surface Areas of Revolution

(See attached file for full problem description) Please follow instructions and show every last step please!

Integral : Average Value of a Function

Find the numbers b such that the average value of f(x) = 2 + 6x - 3x^2 on the interval [0, b] is equal to 3. I know that fave = (1/b-a) (integral sign, b,a) f(x) dx , but I can't figure out how to isolate the b in that equation to find the unknown b.

Integrals : Time Needed to Pump Out a Tank

A vertical right circular cylindrical tank measures 26 ft high and 10 ft in diameter. it is full of oil weighing 60 lbs/ft^3. How long will it take a 1/2 horsepower pump, rated at 275 ft x lbs/sec to pump the oil to the level of the top of the tank? Give your answer to the nearest minute!

Integration Using a Trigonometric Formula : &#8747;&#8730;(1-cos2x)dx

Please evaluate the following integral using basic formulas. Rewrite the integral to match it to a standard formula, and then solve the integral. Then evaluate the answer for the values 0&#8804;x&#8804;pi Please show each step in the problem. Thank you. &#8747;&#8730;(1-cos2x)dx

Viscous Fluid Flow : Viscous Drag on the Walls of a Pipe

For laminar flow in the entrance of to a pipe, as shown in figure, the entrance is uniform u=U0, and the flow downstream is parabolic in profileu(r)=C(r0^2-r^2). Using integrla relations, show that the viscous drag exerted on the pipe walls between 0 and x is... Prescribed text book: White, FM, , Viscous Fluid Flow, 2nd Editi