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# Integrals

### Volume Integrals.

Volume integrals. Calculate the work you do in going from point (1,1) to point (3,3). Choose two different paths, and show that this force field is nonconservative. Please see the attach files.

### Indefinite Integral & Depreciation

Please find the questions in the attachment. 1. Find the indefinite integral and check the result by differentiation: Integrate x^3/ (1 + x^4)^2 dx 2. The rate of depreciation dV/dt of a machine is inversely proportional to the (t + 1)^2 , where V is the value of the machine t years after it was purchased. The initi

### Trapezoid Rule and Simpson's Rule

Please see the file attached. 11. Use the Trapezoid Rule and Simpson's Rule to approximate the value of the definite integral for the given value of . Round your answer to four decimal places and compare the results with the exact value of the definite integral.

### Converting an improper integral

Let f be a non-negative monotone continuous function defined on the interval (0 <). Suppose that the improper integral f(x)dx integrated is convergent. Prove that f is bounded on (0<) and the limit as x goes to infinity of f(x) is 0.

### Force of Newtons and Displacement

A force of 5 Newtons acts in the direction of a = -7i+3j+4k, moving a particle from point B(1, 7, 1) to point C(-7, 5,-2). What is the work done on the particle given that the displacement is measured in meters?

### Singular perturbation theory

Find the expansions of the solutions of x^2 + (4+epsilon) x + 4 - epsilon = 0 around epsilon = 0.

### Evaluating Integrals: Integration by Part

Evaluate the following integral (see the file below). If integration by parts is required, clearly state which functions correspond to u, du, v, and dv. If substitution is required, clearly state the substitution being performed, showing all work.

### Integration By Part

Please see the attachment below. If integration by parts is required, clearly state which functions correspond to u, du, v, and dv. Clearly state the substitution, if required, being performed. Thank you for all of your help in advance!

### Vector field line integral

Consider the vector field F = (x^2 - y^2)i + (y^2 - Z^2)j + (z^2 - x^2)k. a) Calculate the line integral of F from P_0 at (1,0,0) to P_1 at (-1,0,0) along the direct line path joining these points. B) Calculate the line integral of F from P_0 at (1,0,0) to P_1 at (-1,0,0) along the horizontal semicircular path joining these

### Integration by Partial Fraction

Integrate (-6t+14)/(2t^2+9t-5) dt from 0 to 1 What is the approximate value? I could compute (-33t +27)/(2t+5t-3) dt from 0 to 1 by factoring the denominator and express the fraction in terms of its partial fractions, multiply to clear the fraction, solved the pair of equations for A and B, evaluated the indefinite integral

### Properties of Reimann Integral and The Fundamental Theorem

Consider The Following Function: (See Attached File) Calculate the Upper and Lower Reimann sums based on an arbitrary partition P = {xo, .............., xn} of [a,b] and then compute the upper and lower Reimann Integrals of D (5x-11) over [a,b]. Prove that f is not a Reimann Integral.

### Finding the Double Integral

1. Use a double integral to find the area of the region bounded by the graphs of y= x^2 and y= 8-x^2. Provide a sketch and use fubini's theorem to determine the order of integration. 2. Determine the best order of integration to find the double integral ??xe^y^2 da Where the region R is in the first quadrant bounded by the

### Calculations with a Gaussian Quadrature

I need some help completing the following: Find the nodes x_i and the weights w_i so that the Gaussian quadrature of the sum from i=1 to 2 of (w_i) f(x_i) approximating the integral from -1 to 1 of f(x)dx is exact when f(x) is a polynomial of as high degree as possible.

### Riemann integrable functions

Let f be a real function on [a, b]. Suppose that f is Riemann integrable on [c, b] for every a < c < b. (a) Show that if f is also Riemann-integrable on [a, b] then integral b-a(f dx) = limc-a integral b-c(f dx). (b) Give an example of a function g on [a, b] for which limc-a integral b-c(g dx) is defined, while g is n

### Calculate the total charge by evaluating the triple integral

A quarter spherical volume has a charge density f(x,y,z) = 3 sqrt(x^2+y^2+z^2) mico coulombs per cubic meter. Calculate the total charge by evaluating the triple integral (please see the attached file). Do so by changing the spherical coordinates. Don't forget to calculate the Jacobian. ** Please see the attached file for th

### Quadrature (Matlab) Intergral Evaluated Forms

a) Approximate the integral I= integral evaluated from -1 to 1 of f(x) dx with f(x) = 1/(1+x^2) by the composite trapezoidal, midpoint and Simpson's rule using subintervals with lengths h = 1/2, 1/4, 1/8, 1/16, 1/32, 1/64. To this end you may use the analytic result or a very accurate numerical result to compute the error. Plo

### Computation of a Infinite Sum Using Contour Integration

Let Cn denote the positively oriented boundary of the square x = +/- (N + 1/2)pie and y = +/- (N +1/2)pie where N is a positive integer 1) show that int( dz/ (z^2 sin(z))) = i2pie [ 1/6 + 2sum( (-1)^n / (n^2 pie^2)) n=1] 2) show that sum( (-1)^(n + 1) / n^2 = pie^2 / 12

### Maclaurin and Cauchy's Residue

Let C denote the circle |z| = 1, taken counterclockwise, and use the following steps to show that: int(exp(z + 1/z) dz) = i2pie sum(1/ (n!(n + 1)!) 1) By using the Maclaurin series for e^z, write the above integral as sum(1/n! int(z^n exp(1/z) dz) ) 2) Apply Cauchy's residue theorem to evaluate the integrals above.

### Integration by Substitution, Partial Fraction

Integration by Partial Fractions: This is a fascinating method! While there is no general format to follow here, the original integrand must be a rational fraction. Therefore, this is not a method to use in the case of roots in the integrand. The case that will jump out at you quickly is when the integrand has a quadratic in the

### Problems Involving Optimization of Polynomial Integrals

Compute min (a,b,c) the definite integral from -1 to 1 of |x^8 - a - bx - cx^2|^2 dx and find max the definite integral from -1 to 1 of (x^3)(g)(x) dx where g is subject to the restrictions the definite integral from -1 to 1 of g(x) dx = the definite integral from -1 to 1 of xg(x) dx = the definite integral from

### Proof of Estimates

|int(Log(z)/(z^2) dz, Cr| is less than or equal to 2pi((pi + ln(R))/R) where Cr = {z|z = Re^i(theta), theta is an element of [0, 2pi]} and R > 1 In words: show that the following estimates hold The absolute value of the integral of Log(z) over z squared integrated w.r.t z from Cr is less than or equal to 2 pie times pie

### Maximal Ideals of Prime and Primitive Polynomials

Please help with the following problem. Suppose that p is a prime and f is a primitive polynomial in Z[x] which is irreducible modulo p. Show that the set of maximal ideals in Z[x] all can be written as (p,f). Also show that there does not exist a principal ideal in Z[x] which is maximal.

### volume by integration for involving along x-axis

Computer analysis showed that the surface of a certain drumlin can be approximated by y= 10(1-0.0001x^2) revolved 180 about the x-axis from x= -100 to 100. (200m) around. Find the volume of this drumlin (m^3)

### Contour integration problems

Using the method of complex variables and contour rotations, evaluate: Please see the attachment for the questions. Please show the complete steps.

### Calculating an average value of a function

The temprature recorded in a city during a given day approximately followed the curve of T=0.00100t^4-0.280t^2+25 where t is the number of hours from noon (-12hours < t < 12hours). What is the average during the day. a=0, b=5

### Find the length of curve by integration, quadratic equation

Consider that two particles are traveling the paths of the following functions: f(x)=3/4x^2-x+1 g(x)=-x^2+2x+3 If each particle is to begin at one point of intersection at the same time and travel to the other point of intersection (at the same velocities), then which particle would arrive first? WHY?

### Finding the exact area under a curve beween two indicated values of X.

Hello, I would appreciate some assistance with the following problem as I'm not quite grasping the concept. The problem is as follows: Find the exact area under the given curve between the indicated values of X. y=3X, between X=0 and X=3

### Riemand integrable

prove if f(x)=1 if x= 1/n ,n is positive integer f(x)= 0 else, is Riemand integrable on [0,1]

### Differentiate of Functions and Equations

1) Solve the following equation for x. ( five points) a) b) 2) Differentiate the following functions ( five points) a) b) 3) Sketch the following function domain________________ (15 points)