Stewart Calculus · 8th Edition
Chapter 6: Applications of Integration
Every key formula from Stewart Calculus Chapter 6, in one searchable page. Click a card to study it — worked examples included.
6.1 · AREAS BETWEEN CURVES
Top minus bottom (in x)
A = ∫ab [f(x) − g(x)] dx, where f ≥ g
Find where curves cross: solve f(x) = g(x) for a, b.
Right minus left (in y)
A = ∫cd [f(y) − g(y)] dy, where f ≥ g
Use dy when x is easier as a function of y.
WORKED EXAMPLE
Area between y = x² and y = x: cross at x = 0, 1.
A = ∫01 (x − x²) dx = [x²2 − x³3]01 = 16.
A = ∫01 (x − x²) dx = [x²2 − x³3]01 = 16.
6.2 · VOLUMES: DISKS & WASHERS
Disk (solid, no hole)
V = ∫ab π[f(x)]² dx
Washer (hole in the middle)
V = ∫ab π([R(x)]² − [r(x)]²) dx
R = outer radius, r = inner radius.
WORKED EXAMPLE
y = x², 0 ≤ x ≤ 1, about x-axis:
V = ∫01 π(x²)² dx = π∫01 x⁴ dx = π[x⁵5]01 = π5.
V = ∫01 π(x²)² dx = π∫01 x⁴ dx = π[x⁵5]01 = π5.
KEY NOTES
- Axis of rotation = where radius is measured from.
- Rotate y = f(x) about y = k: radius = |f(x) − k|.
6.3 · CYLINDRICAL SHELLS
Shell method (about y-axis)
V = ∫ab 2π(radius)(height) dx
V = ∫ab 2πx · f(x) dx
Unroll the shell: circumference 2πx × height f(x) × thickness dx.
WHICH METHOD?
- Rotate about x-axis, y = f(x) ⇒ disks/washers in dx.
- Rotate about y-axis, y = f(x) ⇒ shells in dx.
WORKED EXAMPLE
y = x, 0 ≤ x ≤ 2, about y-axis:
V = ∫02 2πx · x dx = 2π[x³3]02 = 16π3.
V = ∫02 2πx · x dx = 2π[x³3]02 = 16π3.
6.4 · WORK
Constant force
W = F · d
Variable force
W = ∫ab F(x) dx
Hooke’s Law (springs)
F(x) = kx, W = ∫ab kx dx = 12k(b² − a²)
x measured from natural length; k = spring constant.
WORKED EXAMPLE
Spring k = 40 N/m, stretch 0.3 m:
W = ∫00.3 40x dx = 20(0.3)² = 1.8 J.
W = ∫00.3 40x dx = 20(0.3)² = 1.8 J.
KEY NOTES
- Pumping liquid: W = ∫ ρg · A(y) · d(y) dy (weight × area × lift).
6.5 · AVERAGE VALUE
Average of f on [a, b]
fave = 1b − a ∫ab f(x) dx
The constant height with the same area underneath.
Mean Value Theorem for Integrals
There is c in (a, b) with f(c) = fave
WORKED EXAMPLE
f(x) = x² on [0, 2]:
fave = 12∫02 x² dx = 12 · 83 = 43.
fave = 12∫02 x² dx = 12 · 83 = 43.
VOLUME TOOLKIT
Quick reference
Disk: ∫ πr² dx · Washer: ∫ π(R²−r²) dx
Shell: ∫ 2πxh dx · Cross-section: ∫ A(x) dx
Shell: ∫ 2πxh dx · Cross-section: ∫ A(x) dx
SETUP CHECKLIST
- Sketch the region and the axis of rotation.
- Pick dx or dy: slices ⊥ to the axis.
- Write radius/height in terms of one variable.
WATCH OUT!
Mistakes that cost points
- Washer: square FIRST, then subtract: R² − r² ≠ (R − r)²
- Shell radius is the distance to the axis, not just x
- Work: x starts at the spring’s natural length
- fave divides by (b − a) — don’t forget!
- Area: integrate |f − g| if curves cross
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