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# Integrals

### Integrate the Function

Integrate 6 sqrt x dx.

### Integrate 12/x^4dx

Integrate 12/x^4dx

### Integrate (6x^2-4x^3+5x^4)dx

Integrate (6x^2-4x^3+5x^4)dx.

### Find the area under the curve.

Find the area under the curve y=-x^2+3x from x=1 to x=4

### Continuous Functions, Fundamental Set of Solutions

Consider the attached differential equation where I = (a,b) and p,q are continuous functions on I. (a) Prove that if y1 and y2 both have a maximum at the same point in I, then they can not be a fundamental set of solutions for the attached equation. (b) Let I = {see attachment}. Is {cos t, cos 2t} a fundamental set of solu

### Solve the ODE

Use words to describe the solution process. Typeset solutions. Work is to be done without the aid of a calculator or computer. Show all steps. For example, if you integrate by parts, show all the steps of integration. If you use an integral table, state that as well. Find the solution to y''' + 2y'' + y' = 0 satisfying

### velocity expression satisfies the partial differential equation

Thank you in advance for your help in solving this problem. See attached problem statement.

### Calculating Traction from Stress Tensor Matrix

NOTE: in part A, the traction is just the integral of the dot product of T and n. 7 0 -2 The stress at point P = 0 5 0 -2 0 4 I want to know the traction vector on the plane at point P with the unit normal n = (2i1, -2i2, 1i3)/3

### Numerical Integration: Composite Midpoint Method and Error

Derive the composite midpoint method and composite error.

### Residue Theorem and L'Hopital's Rule

Please evaluate the attached by means of the residue theorem.

### How to Find the Integral of a Function

See attached. Let g(a) be the real solution to x+x^5=a

### Joint Density

The joint denisty function of X and Y is given by [see attached] (a) Find E(X) (b) Find E(Y) (c) Show that Cov(X,Y)=1 *(Please see attachment for complete problem)

### Area Between a Curve

Please assist me with the attached problems relating to finding the region within a curve. 3. (a) Obtain an expretsian far Calculating the area between the curve y=2?x+x2 and the u-axis far 0 <x< 2 by dividing the area up into 2n strips of equal width (each strip will have width 1/n) and then taking the limit as n ---> infini

### Centroid-triple integrals

Find the centroid of the first octant region that is interior to the to the two cylinders x^2+z^2=1 and Y^2+Z^2=1 centroid for x y and z are x'=1/M*triple integral of x^2*dV y'=1/M*triple integral of y^2*dV z'=1/M*triple integral of z^2*dV

### Example of Evaluating integrals

See attached file.

### Evaluate the Integral

&#8747;(2-x)^(3/5) dx Please see the attached file for the fully formatted problems.

### Evaluating an Integral by Substitution

Evaluate the integral in the following: ∫(2-x)^3/5*dx. See the attached file.

### Monte Carlo Integration and Double Integral

Describe how to use the Monte Carlo method to estimate the double integral of xydxdy over the area 0<x<y and 2<y<4. See the attached file.

### Limits : Evaluating Integrals, Anti-Differentiation and Area Between Curves

(a) Consider the attached limit of summed terms (i) Explain why each of the sums in the attached expression gives an over-estimate of the area beneath the curve {see attachment} (ii) Evaluate this limiting sum, using the expression {see attachment} (iii) Check your answer in (ii), by using the fundamental theorem

### Gravitational Attraction : Triple Integral

Find the gravitational attraction of a solid hemisphere of radius a and density 1 on a unit point mass placed at its pole REVIEW: Fz=G*triple integral of density*cos(phi)sin(phi)d(rho)d(phi)d(theta)

(a) Describe how the weights for the order 4 closed Newton-Cotes quadrature formula could be found. Do NOT calculate the weights. (b) What are composite quadrature rules and why are they preferred to using higher order quadrature rules? (c) What are the main characteristics of a predictor-corrector method for solving an initia

### Line Integral : Green's Theorem

A) For what simple closed (positively oriented) curve C in the plane does the line integral of (e^(-x)+ 4x^2y +y)dx + (x^3-x*y^2+5x)dy have the largest positive value? (use Green's theorem) b) Determine what this value is.

### Power Series Expansion and its Utilization in Integration

Problem: Given the power series for the following function (1+x)^k (a) Write the power series for (1+x)^(1/3) (b) Use the power series from part (a) to find the power series for x^3 (c) Using this series approximate the following integral (1+x^3) ^(1/3) using the first three terms

### integral definition of Laplace transform to compute a function

(a) Use the integral definition of the Laplace transform to compute (FUNCTION1) (b) A function g(t) has the transform (FUNCTION2). Use transform properties to compute the following. Express each in simplest form: i) (FUNCTION3) ii) (FUNCTION4) (See attachment for full question).

### First order homogenous differential equation.

Find the explicit solution to the ODE 2yy'=(1+y^2) subject to y(0)=4. What is the solution if y(0)=-4? *(Please see attachment for proper citation of symbols and numbers)

### Prove that the Line Integral of a Vector Field along a Contour C is equal to the Polar Moment of Inertia of the region R lying below C and above the x-axis.

Consider the vector field F=((x^2)*y+(y^3)/3)i,(i is the horizontal unit vector) and let C be the portion of the graph y=f(x) running from (x1,f(x1)) to (x2,f(x2)) (assume that x1<x2, and f takes positive values). Show that the line integral "integral(F.dr)" is equal to the polar moment of inertia of the region R lying below

### Changing variables in multiple integrals

Using the coordinate change u=xy, v=y/x, set up an iterated integral for the polar moment of inertia of the region bounded by the hyperbola xy=1 , the x-axis, and the two lines x=1 and x=2. Choose the order of integration which make the limits simplest THIS MESSAGE IS ADDRESSED TO ANY TA: I found something , I just want you

### Control Systems Under Proportional-Integral Control

Consider a block diagram describing a system under proportional-integral control (as show in figure in attachment): Find the constraints and determine the range (using the Routh-Herwitz criterion) of Kp and Ki. Also, find the closed loop system transfer function assuming the controller gains are set to a specific value. (Ple

### Function and Differential Equations

See attached explanation Differential equations are not my strong suit now. Please explain in a simple way each step from the integral 1/F dF to the final answer. Show and tell how you get from step to step. On problem 35 please answer and explain this in the simplest way you can for me to understand please. Step by

### Functions and Integrals

Please assist me with the attached problems relating to functions and integrals.