Volume by Cross Sections
Not all solids are rotations. Some are built by stacking known shapes (squares, triangles) on top of a 2D base region.
What is the Cross-Section Method?
Calculating volume by Cross Sections is different from the Disk or Shell methods. Instead of spinning a shape around an axis to create a solid, you are "building" a solid on top of a 2D base.
Think of it like a loaf of sliced bread: the base of the loaf sits on the cutting board (the -plane), and each slice has a specific geometric shape (like a square, a semicircle, or a triangle) that gives the loaf its 3D volume.
Volume by Slicing
How it Works: The Geometry
To find the total volume, you find the area of one representative "slice" and then integrate (sum) those areas across the entire length of the base.
The Cross-Section Formula
If cross sections are perpendicular to the -axis, the volume from to is:
Where is the area of the cross-section at any point .
Common Cross-Section Area Formulas
The "side" of your shape (let's call it ) is usually the vertical distance between two functions: .
| Shape | Area Formula |
|---|---|
| Square | |
| Semicircle | |
| Equilateral Triangle | |
| Isosceles Right △ (hyp on base) |
Step-by-Step Process
Worked Example: Square Cross Sections
Volume with Squares
A solid has a circular base: . Cross sections perpendicular to the -axis are squares. Find the volume.
Level Up Examples
Example A: Equilateral Triangles
Same circular base , but cross sections are equilateral triangles.
Example B: Semicircular Cross Sections
Same circular base, but cross sections are semicircles with diameter on the base.
Example C: Base Between Two Curves
The base is the region between and from to . Cross sections are squares.
Practice Quiz
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