Calculus BC · Applications of Integration (Unit 8) · 14 min read · Updated 2026-05-11
Volumes with cross sections: squares and rectangles — AP Calculus BC
AP Calculus BC · Applications of Integration (Unit 8) · 14 min read
1. Cross Sections Perpendicular to the x-axis★★☆☆☆⏱ 4 min
This is the most common case on the AP exam. The base of the solid lies in the $xy$-plane, bounded between two curves $y=f(x)$ and $y=g(x)$ (with $f(x) \geq g(x)$ on $[a,b]$). We slice the solid with vertical planes perpendicular to the x-axis, each slice has thickness $dx$ and cross-sectional area $A(x)$, so total volume is the integral of $A(x)$ over the bounds of the base.
2. Cross Sections Perpendicular to the y-axis★★★☆☆⏱ 4 min
When cross sections are perpendicular to the y-axis, slices are horizontal instead of vertical, so we integrate with respect to $y$. The base is bounded between $x=f(y)$ and $x=g(y)$ (with $f(y) \geq g(y)$ on $[c,d]$), and side length is the horizontal distance between the two curves. If your original curves are given as $y=f(x)$, you must invert them to get $x$ as a function of $y$ first.
3. Special Case Variations★★★☆☆⏱ 3 min
Two common tested variations change how you calculate cross-sectional area from the base side length:
**Squares with diagonal in the base**: If the distance between the base curves equals the diagonal $d$ of the square, use the relationship $d = s\sqrt{2}$, so area $A = \frac{d^2}{2}$.
**Rectangular cross sections**: If the base side is $s$, and height is proportional to $s$ ($h = ks$ for constant $k$), area $A = s \times ks = ks^2$. If height is a constant $h$, area $A = sh$.
4. AP-Style Worked Practice★★★★☆⏱ 3 min
Common Pitfalls
Why: You default to the common side-in-base case without reading the problem carefully
Why: You forget the variable of integration must match the cross section orientation, and skip inverting functions
Why: You subtract top-from-bottom by habit without checking which curve is larger on the interval
Why: You forget area of a rectangle is base × height, skipping the multiplication step