Let R be the region bounded by the graphs of , x = 1, and the x-axis. Draw the solid obtained by rotating R around the x-axis.
Answer
The region R looks like this:
For each problem we need to label the axis of rotation, draw a mirror image of R on the other side of the axis of rotation, then make curvy lines so things appear 3-D. Curvy lines are the technically correct term. If we don't like curvy, we can also use words like winding, snaky or even serpentine. Our goal is to eventually use these images to help us build our integrals.
For each of these, we'll draw R, the axis of rotation, and then the mirror image of R across the axis of rotation. Then we'll make curvy lines to make volumes out of the two areas. We'll see that same region R can be used to make very different solids of rotation just by changing the axis of rotation.
Alright, let's see what this 3-D solid is all about.
Looks kind of like a salad bowl. Who's hungry?
Example 2
Let R be the region bounded by the graphs of , x = 1, and the x-axis. Draw the solid obtained by rotating R around the y-axis.
Answer
The region R looks like this:
Now that we have the region, we need to extend it over the axis of rotation, and then throw in some circles to get a better idea of what this solid is all about.
Hmmm...we're not sure how to describe this solid. A pin cushion perhaps?
Example 3
Let R be the region bounded by the graphs of , x = 1, and the x-axis. Draw the solid obtained by rotating R around the line x = 1.
Answer
The region R looks like this:
Now let's flip this region over the line x = 1, through is a few circles, and see what shape we get.
Hmmm...another salad bowl. We're seeing a recurring theme here.
Example 4
Let R be the region bounded by the graphs of , x = 1, and the x-axis. Draw the solid obtained by rotating R around the line x = 2.
Answer
The region R looks like this:
This time, the axis of rotation is x = 2, which doesn't touch our region. Once we construct our solid, we should expect it to have a hole in the center.
Just like we thought, there's a big cylinder missing from the center. Our salad is going to go right through the bowl and fall on the floor. There goes our lunch plans.
Example 5
Let R be the region bounded by the graphs of , x = 1, and the x-axis. Draw the solid obtained by rotating R around the line x = -1
Answer
The region R looks like this:
Once again, the axis of rotation doesn't border our region, so we should expect our solid to have a chunk missing from it.
This solid is missing a cylinder from its middle, just as we suspected.
Example 6
Let R be the region bounded by the graphs of , x = 1, and the x-axis. Draw the solid obtained by rotating R around the line y = 1.
Answer
The region R looks like this:
This time we'll need to rotate our region around a horizontal line. We'll probably get one of the shapes we got before, just flipped by 90°. Let's find out.
Yep, we've definitely seen this fella before. We're just looking at him sideways now, or cross-eyed.
Example 7
Let R be the region bounded by the graphs of , x = 1, and the x-axis. Draw the solid obtained by rotating R around the line y = 4.
Answer
The region R looks like this:
Just like in the last exercise, we're going to revolve this region around a horizontal line. The difference here is that the line y = 4 doesn't touch our region, so we expect to get a hole in our 3-D solid.
This solid also has a vortex-shaped hole in it, and it's a pretty big one at that. We could probably fit a lot of stuff in there.
Example 8
Let R be the region bounded by the graphs of , x = 1, and the x-axis. Draw the solid obtained by rotating R around the line y = -1.
Answer
The region R looks like this:
We've gotta go below the x-axis for this one, into negative y territory.
The real solid looks like this, with a cylinder missing from the middle.
Now that we've got a feeling for what some of these solids look like, let's start finding their volumes.
, x = 1, and the x-axis. Draw the solid obtained by rotating R around the x-axis.
