Stewart Calculus · 8th Edition
Chapter 8: Further Applications of Integration
Every key formula from Stewart Calculus Chapter 8, in one searchable page. Click a card to study it — worked examples included.
8.1 · ARC LENGTH
L = ∫ab √(1 + [f′(x)]²) dx
Parametric
L = ∫αβ √((dx/dt)² + (dy/dt)²) dt
KEY NOTES
- ds = √(1 + (y′)²) dx is the tiny hypotenuse.
x as a function of y
L = ∫cd √(1 + [g′(y)]²) dy
Use whichever variable gives the simpler derivative.
WORKED EXAMPLE
y = 23(x²+1)3/2, 0 ≤ x ≤ 1:
1 + (y′)² = (2x²+1)² ⇒ L = ∫01(2x²+1)dx = 5/3.
1 + (y′)² = (2x²+1)² ⇒ L = ∫01(2x²+1)dx = 5/3.
y = x³ on [0,1]: L = ∫01 √(1+9x4) dx (use a calculator).
Smooth on [a,b]: f′ continuous ⇒ no corners/cusps.
8.2 · SURFACES OF REVOLUTION
About the x-axis
S = ∫ab 2πy √(1 + (y′)²) dx
About the y-axis
S = ∫cd 2πx √(1 + (x′)²) dy
Radius = distance from the curve to the axis.
About a line y = c
S = ∫ab 2π|y − c| √(1 + (y′)²) dx
ds form
S = ∫ 2π(radius) ds, ds = √(1 + (y′)²) dx
WORKED EXAMPLE
y = x³, 0 ≤ x ≤ 1, about the x-axis:
S = ∫01 2πx³√(1+9x4) dx = (π/27)(10√10 − 1).
S = ∫01 2πx³√(1+9x4) dx = (π/27)(10√10 − 1).
WORKED EXAMPLE
Sphere: y = √(r²−x²), about x-axis:
S = ∫−rr 2πr dx = 4πr².
S = ∫−rr 2πr dx = 4πr².
8.3 · HYDROSTATIC FORCE
F = ∫ab ρg · h(y) · w(y) dy
h = depth below the surface, w = width at depth y.
Water: ρ = 1000 kg/m³, g = 9.8 m/s².
Water: ρ = 1000 kg/m³, g = 9.8 m/s².
KEY NOTES
- Pressure at depth h is P = ρgh, so on a vertical plate F = ρgh̄A with h̄ the centroid’s depth and A the area.
WORKED EXAMPLE
Plate 2 m wide, 3 m tall, top edge at the surface:
F = ∫03 ρg · y · 2 dy = 9ρg = 88,200 N.
F = ∫03 ρg · y · 2 dy = 9ρg = 88,200 N.
8.3 · MOMENTS & CENTER OF MASS
Lamina under y = f(x)
My = ∫ab xρf(x) dx, Mx = 12∫ab ρ[f(x)]² dx
x̄ = My/m, ŷ = Mx/m
WORKED EXAMPLE
Under y = x², 0 ≤ x ≤ 1 (ρ = 1):
m = 1/3, My = 1/4 ⇒ x̄ = 3/4;
Mx = 1/10 ⇒ ŷ = 3/10.
m = 1/3, My = 1/4 ⇒ x̄ = 3/4;
Mx = 1/10 ⇒ ŷ = 3/10.
Pappus’s centroid theorem
V = A · d (solid), S = L · d (surface)
d = distance the centroid travels; the region must not cross the axis.
KEY NOTES
- Region symmetric about the y-axis (uniform ρ) ⇒ x̄ = 0 — only ŷ needs work.
WORKED EXAMPLE
Semicircular lamina y = √(r² − x²): x̄ = 0;
Mx = 2r³/3, m = πr²/2 ⇒ ŷ = 4r/(3π).
Mx = 2r³/3, m = πr²/2 ⇒ ŷ = 4r/(3π).
Mean of X: μ = ∫−∞∞ x f(x) dx.
Centroid of a region: (x̄, ŷ) = (My/A, Mx/A).
8.5 · PROBABILITY
f(x) ≥ 0, ∫−∞∞ f(x) dx = 1
P(a ≤ X ≤ b) = ∫ab f(x) dx
Mean μ = ∫−∞∞ x f(x) dx
KEY NOTES
- Exponential: f(x) = (1/μ)e−x/μ for x ≥ 0.
Variance
σ² = ∫−∞∞ (x − μ)² f(x) dx, SD = σ
Normal distribution
f(x) = 1σ√(2π)e−(x−μ)²/(2σ²)
WORKED EXAMPLE
Exponential with mean μ: P(0 ≤ X ≤ μ)
= [−e−x/μ]0μ = 1 − 1/e ≈ 0.632.
= [−e−x/μ]0μ = 1 − 1/e ≈ 0.632.
WATCH OUT!
Mistakes that cost points
- Arc length: square f′ BEFORE adding 1
- Surface radius is measured to the AXIS
- Hydrostatic depth starts at the SURFACE
- x̄ uses My — they’re crossed!
- A probability density must integrate to 1
- For Mx use ½[f(x)]² — the ½ is easy to forget
- Pappus V = A·d only if the region does NOT cross the axis
No formulas match your search. Try a different keyword.