Expert: Paul Klarreich - 4/2/2008

Question
How do you work these?

1.  What is the average value of y for the part of the curve y= 3x-x^2 which is in the first quadrant?

2. The volume of the solid obtained by revolving the region enclosed by the ellipse x^2+9y^2= 9 about the x-axis is?

3.  Let f and g be odd functions.  If p, r, and s are nonzero functions defined as follows, which must be odd?
         I. p(x)=f(g(x))
         II. r(x)= f(x)+ g(x)
         III. s(x)=f(x)g(x)

4.  The volume of a cylindrical tin can with a top and a bottom is to be 16pi cubic inches.  If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?  

Answer
Questioner:   Dana
Category:  Calculus
Private:  No
 
Subject:  Calculus AP material
Question:  How do you work these?
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Hi, Dana,

Since you didn't indicate what you already tried to do, I shall get you started on them, rather than working them out.

1.  What is the average value of y for the part of the curve y= 3x-x^2 which is in the first quadrant?

A. Find the x-intercept(s).  That gives you the boundaries in the first quadrant.
B. Average value = Integral divided by the width of the interval.
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2. The volume of the solid obtained by revolving the region enclosed by the ellipse x^2+9y^2= 9 about the x-axis is?

A. Take a 'slice': a disk with radius = y = f(x), which you solve for, and thickness (height) = dx.

B. Integrate from 0 to 3.

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3.  Let f and g be odd functions.  If p, r, and s are nonzero functions defined as follows, which must be odd?
        I. p(x)=f(g(x))
        II. r(x)= f(x)+ g(x)
        III. s(x)=f(x)g(x)

For each, compute  p(-x), r(-x), s(-x).  Use the facts:

f(-x) = - f(x),  g(-x) = - g(x).

to see if

p(-x) = - p(x)
r(-x) = - r(x)
s(-x) = - s(x)

4.  The volume of a cylindrical tin can with a top and a bottom is to be 16pi cubic inches.  If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?

Variables are:

h = height of can.
r = radius of can.
A = surface area, TO BE MINIMIZED.


Function:  A = Side rectangle + top + bottom.

A = 2 pi r h  + pi r^2 + pi r^2

Condition:  V = pi r^2 h = 16 pi, which you use to eliminate h.

Eliminate, differentiate, set = 0, solve.