Expert: Paul Klarreich - 3/22/2010

Question
The question is as follows:
Sketch the first quadrant area bounded by y=sqrt(1+x^2) + x, the x-axis, and the y-axis.

(a) Verify that this area is infinite.
(b) The solid obtained by rotating this area around the x-axis is called the Horn of Gabriel. Show that its volume is finite.

I sketched the graph and I am not sure how to work out the bounds. I know that the x/y axis are bounds but I am not sure what to do about the right side of the graph since it is not bounded, do I just use infinity for the bounds and take the integral for the area that way?

For the second part of the question I know I need to take the washer method in order to show that the volume is finite, but again I am stuck trying to figure out how to find the bounds.

Any tips as to how I can find both of the bounds so I can then construct the integrals?

Thank you,
Alex

Answer
Questioner: Alex
Country: United States
Category: Calculus
Private: No
Subject: Calculus II
Sketch the first quadrant area bounded by y=sqrt(1+x^2) + x, the x-axis, and the y-axis.
...........................
(a) Verify that this area is infinite.

>> Not difficult.

If your area is:  lim as B --> infinity of:

{B
| (sqrt(1+x^2) + x) dx
}0

then this is greater than:

{B
| (nothing +  x) dx =
}0

{B
|  x dx = x^2/2 from 0 to B
}0

= B^2/2

and when B -> infinity, so does this.
.................................................
(b) The solid obtained by rotating this area around the x-axis is called the Horn of Gabriel.

NO IT IS NOT, kemosabe. The Gabriel's horn is different.

I believe it is defined by:  

y = 1/x, from x = 1 to infinity.

This graph, when rotated about the x-axis, produces a strange figure:

A. Its surface area is infinite.  (You must integrate  ds, not y dx.)

B. Its volume is finite.

In other words, you can fill it with paint, but you cannot paint it.


Check your vocabulary and let me know what happened.



I sketched the graph and I am not sure how to work out the bounds. I know that the x/y axis are bounds but I am not sure what to do about the right side of the graph since it is not bounded, do I just use infinity for the bounds and take the integral for the area that way?

For the second part of the question I know I need to take the washer method in order to show that the volume is finite, but again I am stuck trying to figure out how to find the bounds.

Any tips as to how I can find both of the bounds so I can then construct the integrals?

Thank you,
Alex