Area, Volume, and Arc Length
Volumes of Solids with Known Cross-Sections
We already mentioned that slicing a solid with known cross-section is like slicing a loaf of cinnamon raisin bread. We could also think of it like we are slicing carrots for pot roast or potato slices for potatoes au gratin. Hungry for more?
When asked to find the volume of one of these solids, we are given a few things to start. We'll be told what the base of the solid is, which way it's being sliced,
and what the slices look like.
We can call this one the stick of butter example. Let R be the region enclosed by the x-axis, the graph y = x2, and the line x = 4. Write an integral expression for the volume of the solid whose base is Rand whose slices perpendicular to the x-axis are squares.
Let's use our attack plan.
1) First we have to understand what the solid is.
The region R looks like this:
Let's turn R on its side to make it easier to think of as the base of the solid.
We're told that if the solid is sliced perpendicular to the x-axis the slices are squares. Since the base of the solid stretches from the x-axis up to the graph y = x2, the side-length of the slice at position x is y = x2. This means if we take slices near x = 0, they'll be tiny. If we take slices near x = 4, they'll be much bigger.
2) Now that we understand what the solid looks like, we need to slice it and find the approximate volume of a slice. We've already sliced it, and we know that each slice is a square with side-length y = x2. Each square has a tiny little bit of thickness Δ x.
To find the volume of the slice we multiply the area of the square by the thickness of the slice to get
(x2)2 Δ x.
3) The variable x goes from 0 to 4 within this solid. When we add the volumes of all the slices and take the limit as the number of slices approaches ∞, we find the volume of the solid is
We recommend drawing pictures. Lots of them. Don't be stingy. You'll use less paper and time drawing a couple extra images than you would getting the wrong answer and having to start all over. At minimum, three of them:
1) the region that forms the base of the solid
2) the region with at least one slice sitting on it
3) a slice all by itself
The best way to get better at these is to practice. Feel free to have your favorite 3-D sweet treat while you go through these exercises.