What is the general formula for the volume of a solid with square cross-sections perpendicular to the x-axis?

The volume is $V = \int_a^b [s(x)]^2 dx$, where $s(x)$ is the side length of the square at any point $x$ and $[a, b]$ are the bounds of the base region.

How do you determine the side length $s(x)$ of a square cross-section if the base is bounded by $f(x)$ and $g(x)$?

The side length $s(x)$ is the vertical distance between the two functions, calculated as $s(x) = f(x) - g(x)$ (assuming $f(x) \geq g(x)$).

What is the primary difference between the volume formula for a square cross-section and a circular cross-section (disk)?

The square cross-section uses $A = s^2$, while the circular cross-section uses $A = \pi r^2$. Squares do not require the $\pi$ constant unless the side length itself contains $\pi$.

When calculating volume with square cross-sections, what does the term $dx$ represent physically?

$dx$ represents the infinitesimal thickness of each square slice. When multiplied by the area $s^2$, it creates an infinitesimal volume $dV$.

What common error occurs when a student confuses a square cross-section with a solid of revolution?

Students often incorrectly add a $\pi$ to the integral or divide the distance between curves by 2 (treating it as a radius) instead of using the full distance as the side length.

If cross-sections are perpendicular to the y-axis, how does the integral setup change?

The integral must be set up with respect to $y$: $V = \int_c^d [s(y)]^2 dy$, where $s(y)$ is the horizontal distance between curves (Right - Left).

Why is it necessary for the area function $A(x)$ to be continuous on the interval $[a, b]$?

Continuity ensures that the integral exists and that the volume is well-defined without gaps or infinite spikes in the cross-sectional area.

How does the volume change if the side length of the square cross-section is doubled?

Since the area is $s^2$, doubling the side length ($2s$) results in four times the area ($(2s)^2 = 4s^2$), thus quadrupling the total volume.

What is the result of integrating $s(x)$ instead of $[s(x)]^2$?

Integrating $s(x)$ calculates the 2D area of the base region, not the 3D volume of the solid.

In a problem where the base is a circle $x^2 + y^2 = r^2$, how do you find the side length $s(x)$ for square cross-sections?

Solve for $y$ to get $y = \pm \sqrt{r^2 - x^2}$. The side length is the distance from the top to the bottom: $s(x) = \sqrt{r^2 - x^2} - (-\sqrt{r^2 - x^2}) = 2\sqrt{r^2 - x^2}$.

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Unit 8: Applications of Integration

Squares as Cross Sections

Summary

Calculating the volume of a solid with square cross-sections involves integrating the area of those squares along a specific axis. By defining the side length of the square as a function of the position along the base, we can accumulate infinitesimal square slices to find the total three-dimensional volume.

1. Definition & Core Concepts

An isometric representation of a solid with a square cross-section rising from a 2D base region.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

Squares vs. Solids of Revolution: Unlike the disk or washer methods, the solid is not formed by spinning a region around an axis. Instead, it is 'built up' vertically from the base.
Side vs. Radius: In square cross-sections, the distance between curves is the entire side of the square. In the disk method, that same distance would represent a radius, requiring a factor of $\pi$ in the formula.

Feature	Square Cross-Section	Disk Method (Revolution)
Area Formula	$A = s^2$	$A = \pi r^2$
Multiplier	None (1)	$\pi$
Shape	Prismatic/Angular	Circular/Cylindrical

5. Exam Strategy & Tips