Expert: Scotto - 4/21/2009

Question
I am somewhat confused, so any help would be greatly appreciated!
For the region bounded by f(x)=(1/x)+2 and g(x)=-x+6 set-up and integral
for the rotation about the line y=-8 and x=-8.

Answer
The two curves intersect.  Set them equal to find where.

That's 2 + 1/x = -x + 6.  

Moving everything to the left we get x - 4 + 1/x = 0.  

Putting the entire equation of x gives (x² - 4x + 1)/x = 0.

According to the quadratic formula, x = [4±√(16-4)]/2.
that's x = (4±√12)/2 = (4±2√3)/2 = 2±√3.

The number 0 is in the interval, so by putting 0 into f and g,
we can see which is higher.  It can easily be seen that
f(0) = 2, g(0) = 6, which means g(x) is the higher curve.

Now to rotate about the line y = -8,
we need to take

⌠2+√3
⌡2-√3 π((g(x)+8)² - (f(x)+8)²)dx

It can be seen that g(x)+8 = -x+14, so (g(x)+8)² = x² - 28x + 196.

It can be seen that f(x)+8 = 1/x + 10,
 so (f(x)+8)² = 1/x² + 20/x + 100.

All you've got left to do now is put both of the functions in the
integral and carry it out.  That's

⌠2+√3
⌡2-√3 π(x² - 28x + 196 - 1/x² - 20/x - 100)dx =

⌠2+√3
⌡2-√3 π(x² - 28x + 96 - 1/x² - 20/x)dx =

π(x^3/3 - 14x² + 96x + 1/x - 20ln(x)) from x=2-√3 to x=2+√3.

Put in the top value of x=2+√3 and subtract off the lower at x=2-√3
to get the answer.  I wouldn't multiply anything by π until I had
subtracted the two values.


Now to rotate about the line x=-8, they equations both need to be
inverted so you have f(y) and g(y).  Note that what I call f(y) is
really the inverse of f(x) and what I call g(y) is really the
inverse of g(x).

Since f(x) = 1/x + 2, say that y = 1/x + 2.
Subtracting 2 from both sides gives us y - 2 = 1/x.
Inverting both sides gives us f(y) = 1/(y-2).

Since g(x) = -x + 6, g(y) = 6 - y.
When x is 1, -1+5=5, then 6-5=1 - check.
When x is 2, -2+6=4, then 6-4=2 - check.
Both functions are lines and agree,
so two points is all we need to check.  

When x is 2-√3, determine y1.
When x is 2+√3, determine y2.
Since it asks us to rotate about y=-8,
⌠y2
⌡y1 π[(g(y)+8)² - (f(y)+8)²] dy.
Determine which function has the higher x value.  
If the function is g, do the following.
If the function is f, reverse f(y) and g(y).