Volumes with known cross-sections

Warm-up

Three quick questions from earlier lessons.

Discover

A loaf of bread is sliced into thin slices. If each slice at position $x$ has area $A(x)$ and thickness $dx$ , what adds up to the loaf's volume?

$\displaystyle\int A(x)\,dx$
The area of the base

Key idea

Volume is the integral of cross-sectional area.

V = \displaystyle\int_a^b A(x)\,dx

, where

A(x)

is the area of the cross-section at

x

. Square of side

s

:

s^2

. Semicircle of diameter

s

:

\tfrac{\pi s^2}{8}

. Equilateral triangle of side

s

:

\tfrac{\sqrt{3}}{4}s^2

.

Worked example

Base: under $y = 2 - x$ for $0 \le x \le 2$ ; square cross-sections.

s = 2 - x

The side at x is the height of the base.

A(x) = (2 - x)^2

Area of the square.

V = \tfrac{8}{3}

−(2 − x)³/3 from 0 to 2.

Your turn, guided

Your turn: base under $y = \sqrt{x}$ , $0 \le x \le 2$ , squares.

V = 2

∫₀² x dx.

Practice ★

A solid's base is the region under $y = \sqrt{x}$ for $0 \le x \le 3$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.
A solid's base is the region under $y = \sqrt{x}$ for $0 \le x \le 4$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.

A solid's base is the triangle under $y = 4x$ for $0 \le x \le 4$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.
A solid's base is the triangle under $y = 1x$ for $0 \le x \le 4$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.

Watch out!

Practice ★★

Same base (under $y = \sqrt{x}$ , $0 \le x \le 2$ ), but the cross-sections are semicircles whose diameter lies in the base. Find the volume.
Same base (under $y = \sqrt{x}$ , $0 \le x \le 4$ ), but the cross-sections are semicircles whose diameter lies in the base. Find the volume.

Exit ticket

Three questions to check you have got it.

A solid's base is the region under $y = \sqrt{x}$ for $0 \le x \le 3$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.
A solid's base is the region under $y = \sqrt{x}$ for $0 \le x \le 4$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.
A solid's base is the region under $y = \sqrt{x}$ for $0 \le x \le 8$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.

Summary card

Known cross-sections

Side $s(x)$ from the base.
Area $A(x)$ of the shape.
$V = \int A(x)\,dx$ .

Watch out: a diameter is twice the radius.

About this lesson