### 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.

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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.

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

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

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

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+≡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≡11(mod 17) using the table in 7. 9.

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

∫2x/(x^2 + 1) dx.

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

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

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

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

Evaluate ∫∫S √(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 ≤ u ≤ 3, 0 ≤ v ≤ 2pi. Please see the attached file for the fully formatted problem.

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..

Evaluate the integers. Please see attached.

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).

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"

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.?

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?

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?

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.

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

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

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.

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

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

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

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]))

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

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))

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