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# Integrals

### Surface Area of Revolution

Find the surface area of revolution generated by revolving the region under the curve y=2x on the interval [0,1] about the x-axis.

### Evaluate the Definite Integrals (Trigonometric and Exponential Integrands) (10 Problems)

A. thrdrt(x^2) + 1 dx B. 4(sec^2)(x) - 5sec(x)tan(x) dx C. (e^2x)/(1+e^2) dx D. (2x + 1)^2 dx E. xe^x^2 dx F. x(1-3x^2)^4 dx G. (tan^3)(x) * (sec^3)(x) dx H. (x^2)(e^-x) dx I. (5 - x)/(2x^2 + x - 1) dx J. 1/(x^3(=) dx

### Differential Equations : Solution and Integrating Factor

Determine if the following equation is exact. If it is, solve it. If not, try to solve it by finding an integrating factor. cosx + y(sinx)y'=0

### Heat Equation with Circular Symmetry : Total Heat Energy, Flow of Heat Energy and Equilibrium Temperature Distribution

8. Heat Equation with Circular Symmetry. Assume that the temperature is circularly symmetric: u u(r,t), where r^2 x^2 | y^2. Consider any circular annulus a ≤ r ≤ b. a) Show that the total heat energy is r π f^b_a cpurdr. b) Show that the flow of heat energy per unit time out of the annulus at r b is: (see attachment

### Congruences, Primitive Roots, Indices and Table of Indices

6. Let g be a primitive root of m. An index of a number a to the base (written ing a) is a number + such that g+&#8801;a(mod m). Given that g is a primitive root modulo m, prove the following... 7. Construct a table of indices of all integers from.... 8. Solve the congruence 9x&#8801;11(mod 17) using the table in 7. 9.

### Integrate by Substitution Problem

∫arctan x/[(x^2)+1] dx. See the attached file for the problem.

### Integrate by Parts

∫2x/(x^2 + 1) dx.

### Fundamental Theorem of Calculus: Integrals and Areas

Find the area of the region 1.) y=x- x^2 points (0,1/4) (1,0) 2.) y=1/x^2 points ((1,1) (2, 1/2) Find and evaluate the integral 1.) integral 1 to 0 2xdx 2.) integral 0 to -1 (2x +1)dx 3.) integral 1 to -1 (2t -1)^2 dt 4.) integral 5 to 2 (-3x +4) dx 5.) integral 4 to 0 1/sq rt 2x +1 dx 6.) integral 2

### Integration Applications : Equation of Solution, Marginal Cost and Maximum Height

1.) You are shown a family of graphs each of which is a general solution of the given differential equation. Find the equation of the particular solution that passes though the indicated point. dy/dx=-5x-2 point (0,2) 2.) Find the cost function for the marginal cost and fixed cost marginal costs fixe

### Integral Test for Convergence

1. Solution. Consider the integral By the Integral Test, we know that converges. Why do we choose 2 NOT 1? Since when we choose 1, then ln1=0. So, 2. Solution. Since , we have We know that diverges. So, by comparison t

### Wave Equation: D'Alembert's Solution

See the attached file. Given that the general solution to the wave equation in one space dimension is given by where f, g are arbitrary twice continuously differentiable functions deduce that the solution s satisfying the initial conditions and for some function v, is (this is a special case of the so called

### Helicoid Integral

Evaluate &#8747;&#8747;S &#8730;(1 + x^2 + y^2) dS where S is the helicoid: r(u,v) = u cos(v)i + u sin(v) j + vk , with 0 &#8804; u &#8804; 3, 0 &#8804; v &#8804; 2pi. Please see the attached file for the fully formatted problem.

### Flux Integrals

Suppose is a radial force field,... is a sphere of radius...centered at the origin, and the flux integral.... Let be a sphere of radius... centered at the origin, and consider the flux integral... . (A) If the magnitude of... is inversely proportional to the square of the distance from the origin,what is the value of..

### Evalute the Integers

Evaluate the integers. Please see attached.

### Area of Surface and Volume of Solid of Revolution

Find the area of the surface generated when the arc of the curve... between t=0 and y=1 is revolved about: a) the y-axis b) the x-axis c) the line y= -1 Please see attached for all twelve questions (circled problems).

### Volume of a Solid of Revolution (2 Problems)

Find the volume of the solid obtained by rotating the region bounded by the given curve about the specific line. (a) y=e^{5x}, y=0, x=0, x=1, about x -axis (b) x=5y-y^2, x=0, about y -axis"

### Average Temperature from a Function

In a certain city the temperature at t hours after 9 A.M. is approximated by the function T(t)=44+12sin(pi(t))/12). What is the average temperature of the city during the period from 9 A.M. to 9 P.M.?

### Area of a Region Between Two Curves,.,.

Consider the region enclosed by the curves 2y=4sqrt{x}, y=5, 2y+4x=8. [Note: the y-axis is not a boundary of this region.] Decide whether to integrate with respect to x or y. What is the area of the region?

### Newton's Law of Cooling : Average Temperature

If a cup of coffee has temperature 95 degrees Celsius in a room where the temperature is 20 degrees Celsius, then according to Newton's Law of Cooling, the temperature of the coffee after t minutes is T(t)=20+75e^(-t/50). What is the average temperature of the coffee during the first half hour?

### Question About Average Value of a Function on an Interval

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.

### Evaluate the integral computed

Compute the integral from 0 ---> infinity. e^(-st)*(1/2)*t^2*e^-t+17*t*e^-tdt

### Evaluate Indefinite and Definite Integrals (9 Problems)

1. Evaluate d/dx (t^3)/(1 + t^2) dt 2. Evaluate d/dx et^2 dt 3. Evaluate the indefinite integral ((t^2)sin(t) + 2sin(t))/(2 + y^2) + e^t dt 4. Evaluate the definite integral sqrt(t)(1 + t) dt 5. Evaluate the indefinite integral (2 - sqrt(x))^r dx 6. Evaluate the indefinite integral (1/x + 1/(x^4) + 1/(x^9) dx 7. Evaluate

### Lebesgue Integral and Monotone Convergence Theorem

Let f be a nonnegative integrable function. Show that the function F defined by F(x)= Integral[from -inf to x of f] is continuous by using the Monotone Convergence Theorem. From Royden's Real Analysis Text, chapter 4. See the attached file.

### Continuous Random Variables : Fubini's Theorem

Show that if {see attachment} is a continuous random variable then ... Please see attachment for complete list of questions.

### Solve Sequences Used in Given Situation

There is a rope that stretches from the top of Maidwell building to a tree on the racecourse, and the length of this rope is 1km. A worm begins to travel along the rope at the rate of 1cm each second in an attempt to get to the other end. then a strange thing happens... some malevolent deity intervenes to make life even hard

### Evaluating integrals

Evaluate the following integrals: (1)The integral of (6sin[2x])/sin(x)dx=____+C (2)The integral of (7-x)(3+[x^2])dx=____+C (3)The integral from 3 to 7 of ([t^6]-[t^2])/(t^4)dt=____+C (4)The integral of (6sin[x])/(1-sin^2[x])dx=____+C

### Evaluating Definite Integrals

Thank you in advance for your help. Evaluate the following integrals: (1)The integral from 1 to 7 of 5/t^4 dt (2) The integral from 0 to 1 of (6+[x]sqrt[x])dx (3)The integral from 7 to 8 of 2^t dt (4)The integral from 0 to 1/2 of 4/(sqrt(1-[x^2]))

### Definite and indefinite integrals

Thank you in advance for your help. Evaluate the following definite and indefinite integrals: (1)The integral of (2x)/([x^2]+1)dx (2)The integral of [(arctan(x))/([x^2]+1)]dx (3)The integral of sqrt([x^3]+[1x^5])dx (4)The integral of (2+x)/([x^2]+1)dx (5)The integral of (2x)/([x^4]+1)dx (6)The integral from 0 to 2 of (x

### Evaluating Integrals via Substitution

Thank you in advance for your help. Evaluate the integrals by making the given substitution: (a)The integral of x(4+x^2)^3dx; (u=4+x^2) (b)The integral of ((sin sqrt[x])/sqrt[3x])dx; (u=sqrt[x]) (c)The integral of e^(3sin(t))cos(t)dt; (u=sin(t))

### General Indefinite Integrals

Thank you in advance for your help. Find the general indefinite integrals: (a)The integral of x(1+2x^2)dx (b)The integral of ((x^2)+1+(2/x^2+1))dx