Expert: Scotto - 1/2/2009

Question
i have couple so if can answer which ever one you like(hope you answer all of them though lol) thanks.

1.Given f(x)=x^(3/2), fin the length of the arc of the function for xE[0,1]

im so sorry for asking so many question its just that these problems are so hard to me. math use to be easy for me and now after i took cal wow i feel stupid. im a senior in high school and trying to survive in this class hope you can help. Thank you.
2. Find the volume by disc around y=1 of y=(x-1)^3, xE[1,3] when the graph is rotated.

3.Use the shell method to find the answer for #2 question.

4.Find the region enclosed between f(x)=x^3-x and g(x)=x/(x^2+1).

5. Use Newton's method to find the real solution to x^3-6x^2+12x-10=0 to 3 decimal places.

6. Given that the area of a rectangle is A. Find the length and width that generate the minimum perimeter for such rectangles.

7. A wooden beam has a rectangular cross section with dimensions h and w. the strength S of the beam is directly proportional to the product of the width and the squre of the height: S=kwh^2. What are the dimensions of the strongest beam that can be cut from a round log with diameter = 24 inches.

8. Use Euler's method to sketch the graph of a solution to the differential equation dy/dx=(Sin(theta))^(1/2). triangle theta=pie/12, thetaE[0,pie/3]. take y(0)=0

Answer
I know the feeling - I took algebra in 8th 1. The length of an are is
⌠
⌡ [1 + (f'(x))²]^0.5 dx

2. The volume is given by (I believe)
⌠3
⌡1 π((x-1)^3-1)² dx.

3. I believe the shell method is to divide the interval from 1 to 3 up into pieces and find the volumn of each of the pieces as if they are a cylinder.

4. Set the two functions equal and solve for x.  It looks like there should be two values at which this occurs.  Evaluate the integral from thw low value to the high value of f(x) - g(x).  This may be the negative of the answer, since it assumes f(x) is on top.  If this is so, the answer is really the integral if g(x) - f(x), which can be found by taking the positive of the negative result.

5. Newton's method is x(n+1) = x(n) - f(x(n)/f'(x(n)).

6. Let the length be L and the width be W.
The area is A=LW, the perimeter is P=2L + 2W.
Solve the first equation in terms of W, put that in the second equation, and find the minimum by differentiating.

7. It is known the them beams height h and and width w fall into the equation h² + w² = D², where D is the diameter of a circle.
Solve this equation for h or w and put that into S.  Note that D is a constnat (24, in this case), sot the equation for S only has one variable in it.  Differentiatie with respect to this variable and set the quantity equal to 0.

8. Euler's method states that y(n+1) = y(n) + hy’(n) + O(h2).
By the way, pie is what you eat.  In math, 3.14159... is pi.
On the computer, Θ is alt-233.  What I'll say about the alt-# when # is near 233 is they're all greek to me ... greek letters, that is.

Question for you: what does Θ have to do with x?