# Reflecting About A Line- Solids

1. May 22, 2013

### Justabeginner

1. The problem statement, all variables and given/known data
Find the volume of the solid generated by revolving the region bounded by the graphs of $y= x^2 - 4x + 5$ and $y= 5- x$ about the line $y= -1$

2. Relevant equations
That is what I'm trying to figure out here, which one of the methods to use.

3. The attempt at a solution

I have these steps so far...

x^2 - 4x + 5= 5 - x
x^2 - 3x = 0
x(x-3)= 0
x= 0 and x= 3
These are my intersection points. From 0 to 3 on this graph, the function 5- x has greater y-values than those of x^2- 4x + 5, so I get (5-x)- (x^2-4x+5)= 3x-x^2. And now I am stumped. What method would I use? And how am I supposed to know if it's a disk, shell, or washer method looking at the graph? I am clueless on how to do that. Thank you for your help.

2. May 22, 2013

### CAF123

Generally it will be easier to do the question via the method of shells or via disks/washers. Boths methods work. I think using disks is easier in this case,

3. May 22, 2013

### Staff: Mentor

You can always use either method (shells vs. disks/washers). The question really is which of the two methods results in an integral that is easier to evaluate.

I find it helpful in these problems to draw two graphs. The first shows the region that is being revolved; the second shows a cross-section of the solid of revolution, with some detail on the typical volume element (a shell or disk/washer), including its dimensions.

4. May 22, 2013

### Justabeginner

So using the disk method which if I'm not mistaken is (f(x)^2) dx, I should get:

(3x-x^2)^2= 9x^2 - 6x^3 + x^4 with a and b being 0 and 3 respectively.

108- [0]= 108?

But when I draw the graph.. I don't see how the y= -1 would affect this at all then?

5. May 22, 2013

### LCKurtz

Which means it's wrong, eh? The formula$$\int_a^b \pi f^2(x)dx$$is for the area under $y=f(x)$ rotated about the $x$ axis. That isn't what you have, and you forgot the $\pi$ anyway. You want the formula$$V=\int_a^b \pi (r_{upper}^2 - r_{lower}^2)\, dx$$where $r$ is the radius of revolution for the appropriate curve. The axis being $y=-1$ will affect that.

6. May 22, 2013

### Justabeginner

So by using the correct formula, I get R = (5-x)+1 and r = (x^2-4x+5)+1

Then after I plug in and solve, V= 162pi/5. Is this correct?

7. May 22, 2013

### Staff: Mentor

It would be helpful if you showed the work you did to get that result. I'm not saying that it's wrong, but I can't tell without duplicating your effort.

8. May 22, 2013

### Justabeginner

Well I did this:

pi * integral { (5-x)+1)^2) with a=0 and b=2 - ((x^2-4x+6)^2)} dx

Then I just simplified.

9. May 22, 2013

### Staff: Mentor

This looks good so far.

$$\pi \int_0^3 [(6 - x)^2 - (x^2 - 4x + 6)^2]dx$$