Expert: Paul Klarreich - 12/12/2006

Question
maybe you help with just these 3 questions?

http://img517.imageshack.us/img517/6128/untitled2im6.jpg
http://img470.imageshack.us/img470/9630/untitled5dr4.jpg

Answer
Questioner:  paul
Category:  Calculus
 
Question:  maybe you help with just these 3 questions?

http://img517.imageshack.us/img517/6128/untitled2im6.jpg
http://img470.imageshack.us/img470/9630/untitled5dr4.jpg
....................................
Hi, Paul,

On your first one, #13, you want the area between y = x^2 and y = 2 + x.

First: Find their intersection points:  Solve  x^2 = 2 + x

x^2 - x - 2 = 0
(x - 2)(x + 1) = 0
x = 2, x = -1.
Those are your boundaries of integration.  You will integrate the difference between the functions.  On this interval, the line is higher than the parabola.

{2
| [2 + x - x^2] dx
}-1

= 2x + x^2/2 - x^3/3  from -1 to 2:

( 2(2) + (2)^2/2 - (2)^3/3) - ( 2(-1) + (-1)^2/2 - (-1)^3/3 )

= (4 + 2 - 8/3) - ( -2 + 1/2 + 1/3 )
= 6 - 8/3 + 2 - 1/2 - 1/3
= 8 - 9/3 - 1/2
= 5 - 1/2 = 4 and a half.
...............................................

On the second, #14, the control gate problem, I have no idea what they are talking about -- sorry.

.............................................

On your #15, the 'ring', this looks as if it can be handled by the method of disks.  A 'sample' disk will be a ring with:

Outer radius = 6
Inner radius = 6 - (4x - x^2) = 6 - 4x + x^2
Thickness = dx

The area of a ring (annulus) is  pi(r2^2 - r1^2)

The volume element will be:

dV = pi[ 6^2 - (6 - 4x + x^2)^2 ] dx

The boundaries will be x = 0 to x = 4.  Integrate:

{
| pi[ 36 - (36 - 48x + 12x^2 + 16x^2 - 8x^3 + x^4) ] dx
}

{
| pi[ 36 - 36 + 48x - 28x^2 + 8x^3 - x^4) ] dx
}

{
| pi[ 48x - 28x^2 + 8x^3 - x^4 ] dx
}


= pi[ 24x^2 - 28x^3/3 + 2x^4 - x^5/5) ]  from x = 0 to x = 4

= pi[ 24(16) - 28(64)/3 + 2(256) - 1024/5 ]

= 128 pi[ 3 - 14/3 + 2(2) - 8/5 ]
= 128 pi[ 7 - 14/3 - 8/5 ]
         105     70     24
= 128pi [ ---- - ---- - ---- ]
          15     15     15

         11
= 128pi [ --- ]
         15

    1408
= pi ----
     15

= (93 and 13/15)pi