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Not every interesting solid is a solid of revolution. Some solids have cross-sections that are squares, triangles, semicircles, or other shapes. The slicing principle still works—we just need to express the cross-sectional area in terms of the position variable.
The key is that the base of each cross-section often comes from a curve in the $xy$-plane, and the shape of the cross-section is given in the problem.
The key insight: Identify how the cross-section's dimensions relate to the base curve. Usually, one dimension of the cross-section equals the width of the region at that point.
Self-check: Can you answer these questions? If not, review the linked prerequisites first.
| Question | If you struggle... |
|---|---|
| Explain why $V = \int_a^b A(x)\,dx$ gives volume | Review Volume by Slicing |
| Find the area of an equilateral triangle with side 4 | You need: $A = \frac{\sqrt{3}}{4}s^2 = 4\sqrt{3}$ |
| If $(x-1)^2 + y^2 = 4$, express $y$ in terms of $x$ | Rearranging equations to find cross-section dimensions is essential |
| Shape | Area Formula | Example |
|---|---|---|
| Square (side $s$) | $s^2$ | Side 3 → Area = 9 |
| Equilateral triangle (side $s$) | $\frac{\sqrt{3}}{4}s^2$ | Side 4 → Area = $4\sqrt{3}$ |
| Isosceles right triangle (legs $\ell$) | $\frac{1}{2}\ell^2$ | Legs 2 → Area = 2 |
| Isosceles right triangle (hypotenuse $h$) | $\frac{h^2}{4}$ | Hypotenuse 4 → Area = 4 |
| Semicircle (diameter $d$) | $\frac{\pi d^2}{8}$ | Diameter 4 → Area = $2\pi$ |
Tip: Memorize these—they appear constantly in known cross-section problems.
The "width" of the base region at position $x$ usually becomes the key dimension of each cross-section.
For a region between two curves $y = f(x)$ and $y = g(x)$: $$\text{width} = f(x) - g(x) \quad \text{(assuming } f(x) \geq g(x)\text{)}$$
For a circular base $x^2 + y^2 = r^2$: $$\text{width} = 2y = 2\sqrt{r^2 - x^2}$$
This width becomes the side of a square, the base of a triangle, or the diameter of a semicircle.
| Property | Value |
|---|---|
| Concept | Applications of Integration |
| Course | MATH162 |
| Section | Stewart 5.2 |
| Difficulty | Advanced |
| Time | ~25 minutes |
A solid has a known base (usually a region in the $xy$-plane) and cross-sections of a specified shape perpendicular to one axis.
y
| _____
| / \ ← Base region in xy-plane
| / \
| / ▲ \ ← Cross-sections (triangles, squares, etc.)
| / | \ stand perpendicular to the plane
+---------------→ x
3D view:
/\
/ \
/ \ ← Triangular cross-section
/______\
/ \
/__________\ ← Base$$\boxed{V = \int_a^b A(x)\,dx}$$
where $A(x)$ is the area of the cross-section at position $x$, expressed using the shape's area formula.
| Shape | Area Formula | If base/diameter = $w$ |
|---|---|---|
| Square | $s^2$ | $A = w^2$ |
| Equilateral Triangle | $\frac{\sqrt{3}}{4}s^2$ | $A = \frac{\sqrt{3}}{4}w^2$ |
| Isosceles Right Triangle | $\frac{1}{2}(\text{leg})^2$ | $A = \frac{1}{2}w^2$ (if legs = $w$) |
| Semicircle | $\frac{1}{2}\pi r^2$ | $A = \frac{\pi}{8}w^2$ (diameter = $w$) |
| Circle | $\pi r^2$ | $A = \frac{\pi}{4}w^2$ (diameter = $w$) |
Key question: What determines the size of each cross-section?
Usually, the "width" of the base region at position $x$ becomes a dimension of the cross-section:
$$\text{width at } x = f(x) - g(x) \text{ (top curve minus bottom curve)}$$
or
$$\text{width at } x = 2y = 2\sqrt{\text{something}}$$
The base of a solid is a triangular region with vertices at $(0, 0)$, $(1, 0)$, and $(0, 1)$. Cross-sections perpendicular to the $x$-axis are squares. Find the volume.
A solid has a circular base of radius 2 (the disk $x^2 + y^2 \leq 4$). Cross-sections perpendicular to the $x$-axis are equilateral triangles with one side in the base. Find the volume.
The base of a solid is the region bounded by $y = x^2$ and $y = 1$. Cross-sections perpendicular to the $y$-axis are semicircles with diameters in the base. Find the volume.
The base of a solid is the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with the hypotenuse in the base. Find the volume.
A decorative pillar has a square base with side length 2 meters. The pillar tapers upward with square cross-sections. If the cross-section at height $h$ (measured in meters from the base) has side length $s(h)$, and the pillar has total height $H$:
The Variable Cookie Cutter: Imagine a cookie cutter that changes shape (or size) as you move along the solid. At each position, it cuts a cross-section of a specific shape. The volume is built up by stacking infinitely many of these cookies, each with area $A(x)$ and thickness $dx$.
| Shape | Area | Notes |
|---|---|---|
| Square (side $s$) | $s^2$ | |
| Equilateral triangle (side $s$) | $\frac{\sqrt{3}}{4}s^2$ | Height = $\frac{\sqrt{3}}{2}s$ |
| Isosceles right (legs $\ell$) | $\frac{1}{2}\ell^2$ | Hypotenuse = $\ell\sqrt{2}$ |
| Isosceles right (hypotenuse $h$) | $\frac{h^2}{4}$ | Legs = $\frac{h}{\sqrt{2}}$ |
| Semicircle (diameter $d$) | $\frac{\pi d^2}{8}$ | Radius = $\frac{d}{2}$ |
| Circle (diameter $d$) | $\frac{\pi d^2}{4}$ | Radius = $\frac{d}{2}$ |
Looking back:
Looking ahead:
Real-world applications:
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|---|---|---|
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Last updated: 2026-01-23