Volumes with cross sections: squares and rectangles

AP Calculus BC· AP Calculus BC CED — Applications of Integration· 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 $x y$ -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 $d x$ and cross-sectional area $A (x)$ , so total volume is the integral of $A (x)$ over the bounds of the base.

📘 Definition

General Volume Formula (x-perpendicular cross sections)

V = \int_a^b A(x) dx

Total volume equals the integral of cross-sectional area from the left bound $a$ to right bound $b$ of the base region.

📐 Worked Example

The base of a solid is the region bounded by $y = x^{2}$ and $y = 4$ on $[0, 2]$ . Cross sections perpendicular to the x-axis are squares with one side lying in the base region. Find the volume of the solid.

1
Identify bounds and side length function (vertical distance between curves):
2
$a=0, \quad b=2, \quad s(x) = 4 - x^2$
3
Calculate cross-sectional area for a square with side $s (x)$ :
4
$A(x) = (4 - x^2)^2 = 16 - 8x^2 + x^4$
5
Set up and evaluate the integral using the Fundamental Theorem of Calculus:
6
$V = \int_0^2 (16 - 8x^2 + x^4) dx = \left[16x - \frac{8}{3}x^3 + \frac{1}{5}x^5\right]_0^2 = \frac{256}{15} \approx 17.07$

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.

📘 Definition

General Volume Formula (y-perpendicular cross sections)

V = \int_c^d A(y) dy

Total volume equals the integral of cross-sectional area from the lower bound $c$ to upper bound $d$ of the base region along the y-axis.

📐 Worked Example

The base of a solid is the region bounded by $y = x$ , the x-axis, and $x = 4$ . Cross sections perpendicular to the y-axis are squares with one side lying in the base. Find the volume of the solid.

1
Rewrite curves as functions of $y$ , find bounds:
2
$y = \sqrt{x} \implies x = y^2, \quad c=0, \quad d=2$
3
Calculate side length (horizontal distance between curves):
4
$s(y) = 4 - y^2$
5
Calculate cross-sectional area and integrate:
6
$A(y) = (4 - y^2)^2, \quad V = \int_0^2 (16 - 8y^2 + y^4) dy = \frac{256}{15} \approx 17.07$

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 $d = s 2$ , so area $A = \frac{d ^{2}}{2}$ .
Rectangular cross sections: If the base side is $s$ , and height is proportional to $s$ ( $h = k s$ for constant $k$ ), area $A = s \times k s = k s^{2}$ . If height is a constant $h$ , area $A = s h$ .

📐 Worked Example

The base of a solid is the region bounded by $y = x$ and $y = x^{2}$ on $[0, 1]$ . Cross sections perpendicular to the x-axis are squares whose diagonal lies in the base region. Find the volume of the solid.

1
Find diagonal length at any $x$ :
2
$d(x) = x - x^2$
3
Calculate cross-sectional area for a square with diagonal $d (x)$ :
4
$A(x) = \frac{d(x)^2}{2} = \frac{(x - x^2)^2}{2}$
5
Set up and evaluate the integral:
6
$V = \int_0^1 \frac{x^2 - 2x^3 + x^4}{2} dx = \frac{1}{60}$

4. AP-Style Worked Practice★★★★☆⏱ 3 min

📐 Worked Example

The base of a solid is the region bounded by $y = e^{x}$ , $y = 0$ , $x = 0$ , and $x = 1$ . Cross sections perpendicular to the x-axis are rectangles where the height of each rectangle is equal to 3 times the length of its base in the xy-plane. What is the volume of the solid?

1
Base side length at any $x$ : $s (x) = e^{x}$ , height $h = 3 e^{x}$
2
Cross-sectional area:
3
$A(x) = e^x \times 3e^x = 3e^{2x}$
4
Integrate to find volume:
5
$V = \int_0^1 3e^{2x} dx = \frac{3}{2}(e^2 - 1)$

5. Common Pitfalls

Wrong move:

When the diagonal of the square is in the base, calculate area as $d^{2}$ instead of $d^{2} /2$

Why:

You default to the common side-in-base case without reading the problem carefully

Correct move:

Underline whether the segment in the base is the side or diagonal before starting your calculation

Wrong move:

For cross sections perpendicular to the y-axis, leave side length in terms of x and integrate with respect to y

Why:

You forget the variable of integration must match the cross section orientation, and skip inverting functions

Correct move:

Write all functions and side lengths in terms of the axis variable (x for x-perpendicular, y for y-perpendicular) before setting up the integral

Wrong move:

Subtract curves in the wrong order, get a negative side length, and forget to take absolute value

Why:

You subtract top-from-bottom by habit without checking which curve is larger on the interval

Correct move:

Confirm your side length is positive for all points in the integration interval, swap subtraction order if needed

Wrong move:

For a rectangle with height proportional to base side, use area $= k s$ instead of $k s^{2}$

Why:

You forget area of a rectangle is base × height, skipping the multiplication step

Correct move:

Explicitly write out $base = s, height = k s, Area = s \times k s$ before proceeding

Wrong move:

When squaring $(a - b x^{n})$ , expand to $a^{2} - b^{2} x^{2 n}$ skipping the middle term

Why:

You rush the algebra step before integrating

Correct move:

Always expand the square of a binomial term-by-term to avoid missing the cross term $- 2 ab x^{n}$

6. Quick Reference Cheatsheet

Category	Formula	Notes
Volume (perp to x-axis)	$V = \int_{a}^{b} A (x) d x$	$a, b$ = x-bounds of base, $A (x)$ = cross-section area
Volume (perp to y-axis)	$V = \int_{c}^{d} A (y) d y$	$c, d$ = y-bounds of base, $A (y)$ = cross-section area
Square (side in base)	$A = s^{2}$	$s$ = distance between base boundary curves
Square (diagonal in base)	$A = \frac{s ^{2}}{2}$	$s$ = diagonal length = distance between boundaries
Rectangle (base $s$ , height $h$ )	$A = s \cdot h$	If $h = k s$ , $A = k s^{2}$ ; if $h$ constant, $A = h s$
Vertical side length (x-perp)	$s (x) = f (x) - g (x), f \geq g$	Between two y-boundary curves
Horizontal side length (y-perp)	$s (y) = f (y) - g (y), f \geq g$	Between two x-boundary curves

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · MCQ
Volume with square cross sections
2019 · FRQ
Cross sections perpendicular to y-axis

What's Next

This topic is the foundational introduction to the slicing method for volume, which you will immediately extend to other cross-sectional shapes and volumes of revolution next. Volumes of revolution (disks/washers and cylindrical shells) are just special cases of the slicing method: disks are circular cross sections, so you replace the square/rectangle area formula with the area of a circle. Without mastering the setup of cross-sectional volume integrals, it is very easy to mix up variables and formulas for volumes of revolution, which are a larger tested portion of the AP Calculus BC exam. This topic also reinforces your core skill of finding the distance between two curves, which is used in nearly all applications of integration.