Volume of a solid with a known cross section is a fundamental concept in integral calculus that allows us to compute the three‑dimensional size of an object when we know the shape and area of its slices perpendicular to a chosen axis. By integrating the area of these cross‑sections along the axis, we can determine the total volume without needing a explicit formula for the whole solid. This method is especially useful for irregular shapes, solids of revolution with non‑circular slices, and many engineering applications where cross‑sectional data are readily available from measurements or drawings That's the part that actually makes a difference. Practical, not theoretical..
Introduction
When faced with a solid whose boundary is difficult to describe analytically, the cross‑sectional (or “slice”) method provides a powerful alternative. Imagine stacking infinitesimally thin plates, each having a known area (A(x)) that varies with position (x) along an axis. The volume (V) is then the sum of the volumes of all these plates:
[ V = \int_{a}^{b} A(x),dx ]
where ([a,b]) is the interval over which the solid extends. The key to applying this formula lies in correctly expressing (A(x)) in terms of the variable of integration.
Understanding the Cross‑Sectional Method
What Is a Cross Section?
A cross section is the intersection of a solid with a plane. If the cutting plane is perpendicular to a chosen coordinate axis (usually the (x)-axis), the resulting slice is a two‑dimensional shape whose area can be expressed as a function of the coordinate where the cut occurs.
Why Does Integration Work?
Each slice has thickness (dx) and area (A(x)). Consider this: its volume is approximately (A(x),dx). Adding (integrating) these infinitesimal volumes from the leftmost to the rightmost slice yields the exact volume in the limit as (dx\to0).
Choosing the Axis of Integration
The axis should be parallel to the direction in which the cross‑sectional area changes in a simple, describable way. Often the problem statement hints at this axis (e.g.Practically speaking, , “cross sections perpendicular to the (x)-axis”). If the solid is symmetric, any axis through the center of symmetry works, but the algebra may be simpler in one direction than another Turns out it matters..
Setting Up the Integral
- Identify the variable (x) (or (y)) that runs along the chosen axis.
- Determine the limits (a) and (b): the smallest and largest values of the variable where the solid exists.
- Express the cross‑sectional area (A(x)) as a function of (x). This may involve geometry formulas (area of a rectangle, triangle, circle, etc.) where the dimensions themselves are functions of (x).
- Write the volume integral (V=\int_{a}^{b}A(x),dx).
- Evaluate the integral analytically or numerically, depending on the complexity of (A(x)).
Common Cross‑Sectional Shapes and Their Area Formulas
| Shape | Area Formula | Typical Dependence on (x) |
|---|---|---|
| Rectangle | (A = \text{width} \times \text{height}) | width or height may be linear functions of (x) |
| Triangle | (A = \frac{1}{2} \times \text{base} \times \text{height}) | base and/or height vary with (x) |
| Semicircle | (A = \frac{1}{2}\pi r^{2}) | radius (r) is a function of (x) |
| Equilateral triangle | (A = \frac{\sqrt{3}}{4}s^{2}) | side length (s) varies with (x) |
| Washer (annulus) | (A = \pi(R^{2}-r^{2})) | outer radius (R) and inner radius (r) depend on (x) |
| General polygon | Decompose into simpler shapes | each component’s dimensions are functions of (x) |
Note: When the cross section is a washer, the solid often results from subtracting one solid from another (e.g., a hollow tube).
Step‑by‑Step Procedure (with a Checklist)
- [ ] Sketch the solid and indicate the axis of integration.
- [ ] Draw a typical cross section perpendicular to that axis.
- [ ] Label all dimensions that change with the integration variable.
- [ ] Write each dimension as an explicit function of the variable (use given equations, similar triangles, or geometric relationships).
- [ ] Substitute these functions into the appropriate area formula to obtain (A(x)).
- [ ] Determine the integration limits by finding where the solid starts and ends along the axis.
- [ ] Set up the integral (V=\int_{a}^{b}A(x),dx).
- [ ] Simplify the integrand if possible (expand, factor, combine like terms).
- [ ] Integrate using basic antiderivative rules or substitution.
- [ ] Evaluate the definite integral and include units (e.g., (\text{cm}^{3})).
Worked Examples
Example 1: Solid with Square Cross Sections
Problem: The base of a solid lies in the region bounded by (y = x^{2}) and (y = 4) in the (xy)-plane. Cross sections perpendicular to the (x)-axis are squares. Find the volume.
Solution:
- The solid extends from where the curves intersect: solve (x^{2}=4\Rightarrow x=\pm2). So (a=-2), (b=2).
- At a given (x), the side length of the square equals the vertical distance between the curves:
[ s(x)=4 - x^{2} ] - Area of a square: (A(x)=s^{2}=(4 - x^{2})^{2}).
- Volume integral:
[ V = \int_{-2}^{2} (4 - x^{2})^{2},dx ] - Expand: ((4 - x^{2})^{2}=16 - 8x^{2}+x^{4}).
- Integrate:
[ V = \left[16x - \frac{8}{3}x^{3} + \frac{1}{5}x^{5}\right]_{-2}^{2} ] - Evaluate (using symmetry, compute from 0 to 2 and double):
[ V = 2\left(16(2) - \frac{8}{3}(2)^{3} + \frac{1}{5}(2)^{5}\right) = 2\left(32 - \frac{64}{3} + \frac{32}{5}\right) ]
[ V = 2\left(\frac{480 - 320 + 96}{15}\
