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I have to find the volume of a solid using the disk/washer method and the shell method.  I think I have the first part(disk/washer) right.   I think the shell is off. The problem is "the region in the first quadrant that is bounded above by the curve y = 1/x^1/4, on the left by the line x = 1/16, and below bythe line y = 1 is revolved about the x-axis to generate a solid."    I am having computer problems so any help is appreciated. Thanks

A 37 foot ladder is placed against a wall that is 9 feet away from its base. Will the top of the ladder reach a window ledge that is 35 feet above ground? Explain.

True of False, Explain:

If ∏/2<θ<∏, Then cos θ/2<0

The problem is The square root of 16-x^2 over the interval [0,-4]  0 being the upper bound, -4 being the lower bound.  I have solved 3/4s of this problem but I don't understand what they mean by "Solve the definite integral exactly by geometry". 

Is there a way to do the following on Maple:

I want Maple to use Jacobi's method to give an approximation of the solution to the following linear system, with a tolerance of 10^(-2) and with a maximum iteration count of 300.


The linear system is






Egor has two parents, four grandparents, and so on.
Write an explicit formula and a recursive formula for the number of ancestors Egor has if we go back n generations.                

what would be the figure back to 25 generations ?

Let (G, ·) be a group and X any set. Let F be the set of functions with domain

X and range G. Define a binary operation ∗ on F by (f ∗ g)(x) := f(x) · g(x). Is

prove that this is so.

Yes, (F, ∗) is a group.

prove it.

Exercise Prove that (-1)u = - u in any vector space. Note that (-1)u means the number -1 is multiplied to the vector u, and - u means the negative vector in the fourth property of the definition of vector spaces.


Exercise Prove that (a1u1 + a2u2) + (b1u1 + b2u2) = (a1 + b1)u1 + (a2 + b2)u2 in any vector space.


Exercise Give a detailed reason why, in any vector space,

  • u + v = 0 ⇒ u = - v.

  • 3u + 2v - 4w = 0 ⇒ v = - 3/2 u + 2w.

Solve, using 4000 miles for the radius of the earth.                                                                                              


A space shuttle is in circular orbit 150 miles above the surface of the earth. Approximate                                                                             

  1. the speed
  2. the time required for one revolution.



Suppose an ideal gas expands to four times its initial volume. From experience for this process, the initial and final 

temperature are the same.

  1. Using a macroscopic approach, calculate the entropy change for the gas
  2. Using statistical considerations, calculate the change in entropy for the gas and show that it agrees with the answer 
    you obtained in part 1.

What is the probability that the total of two dice will be greater than 9, given that the first die is a 5?


  1. Sketch the two vectors listed after the formula for r (t).
  2. Sketch, on the same plane, the curve C dterminated by r (t), and indicate the orientation for the given values of t.

    r (t)=t*i +4*cos (t)j +9*sin (t)k,

    r (0), r(Pi/4),  t>=0

How to convert an  algebraic equation to a parametrized one ?


How to sketch the curve of a circular cylinder , x^2+y^3^=a^2 with the parametric equations lying on it ?

x=a*cos't) , y=a*sin(t) , z=b*t , if t varies from 0 to 2*Pi, the point P starts at(a,0,0) and moves upward.

How to show that the parametric equations are indeed lying on the cylinder ?

How to plot the whole things ?


A concho-spiral is a curve C that has a parametrization :





where a, b, mu, are constants.

  1. Show that C lies on the cone a^2*z^2=b^2*(x^2+y^2).
  2. Sketch C for a = b = 4 and mu=-1.
  3. Find the length of C corresponding to the t-interval [0,infinity].

 1) find  the direction in which f(x,y) increases most rapidly at the point P(1,2), and find the maximum rate of increase
of f at P.

Interpret 1) using the graph of f.


Let f(x,y) = 2+x^2 +1/4*(y^2)

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