Stewart Calculus · 8th Edition
Chapter 15: Multiple Integrals
Every key formula from Stewart Calculus Chapter 15, in one searchable page. Click a card to study it — worked examples included.
15.1–15.2 · DOUBLE INTEGRALS
∫∫R f(x,y) dA = ∫ab∫cd f(x,y) dy dx
Type I: y between curves · Type II: x between curves.
WORKED EXAMPLE
∫∫[0,1]×[0,2] (x + y) dA:
∫02 (1/2 + y) dy = 3.
∫02 (1/2 + y) dy = 3.
Fubini’s Theorem
f continuous on rectangle R ⇒ ∫∫R f dA = either iterated integral
Order never matters on a rectangle.
WORKED EXAMPLE
∫∫[0,2]×[1,3] xy dA = (∫02x dx)(∫13y dy) = 2 · 4 = 8.
15.3 · POLAR COORDINATES
∫∫D f dA = ∫αβ∫r₁r₂ f(r cosθ, r sinθ) r dr dθ
KEY NOTES
- The extra r is the Jacobian — never skip it!
WORKED EXAMPLE
Volume under z = 4 − x² − y²:
2π∫02 (4 − r²)r dr = 8π.
2π∫02 (4 − r²)r dr = 8π.
Polar area
A = ∫∫D dA = ∫αβ∫0r(θ) r dr dθ
WORKED EXAMPLE
Area of the disk r ≤ 3:
∫02π∫03 r dr dθ = 2π · 9/2 = 9π.
∫02π∫03 r dr dθ = 2π · 9/2 = 9π.
15.4 · APPLICATIONS
Mass m = ∫∫D ρ dA, x̄ = (1/m)∫∫ xρ dA
Surface area
S = ∫∫D √(1 + fx² + fy²) dA
Surface area
S = ∫∫D √(1 + fx² + fy²) dA
Area of the graph z = f(x,y) over D.
Moments & center of mass
My = ∫∫ xρ dA, Mx = ∫∫ yρ dA; (x̄, ȳ) = (My/m, Mx/m)
Moment of inertia
Ix = ∫∫ y²ρ dA, Iy = ∫∫ x²ρ dA, I₀ = Ix + Iy
Probability
P((X,Y) ∈ D) = ∫∫D f dA; f ≥ 0, ∫∫ f = 1
WORKED EXAMPLE
ρ = x on [0,1]×[0,1]:
m = ∫01∫01 x dy dx = ∫01 x dx = 1/2.
m = ∫01∫01 x dy dx = ∫01 x dx = 1/2.
15.6–15.7 · TRIPLE & CYLINDRICAL
∫∫∫E f(x,y,z) dV as an iterated integral
Cylindrical: x = r cosθ, y = r sinθ, z = z;
dV = r dz dr dθ.
dV = r dz dr dθ.
Volume & average value
V = ∫∫∫E dV · f̄ = 1V(E)∫∫∫E f dV
Symmetry shortcut
E symmetric and f odd in one variable ⇒ integral = 0.
WORKED EXAMPLE
Volume under z = 9 − r² above the xy-plane:
2π∫03 (9 − r²)r dr = 2π · 81/4 = 81π/2.
2π∫03 (9 − r²)r dr = 2π · 81/4 = 81π/2.
KEY NOTES
- Pick coordinates from the REGION’s shape: sphere ⇒ spherical, cylinder / axis of symmetry ⇒ cylindrical, box ⇒ rectangular.
- Sketch E first — the limits follow the picture.
15.8–15.9 · SPHERICAL & CHANGE
x = ρ sinφ cosθ, y = ρ sinφ sinθ, z = ρ cosφ
dV = ρ² sinφ dρ dφ dθ
Change of variables: multiply by |∂(x,y)/∂(u,v)|.
WORKED EXAMPLE
Volume of the unit ball:
∫02π∫0π∫01 ρ² sinφ dρ dφ dθ = 4π/3.
∫02π∫0π∫01 ρ² sinφ dρ dφ dθ = 4π/3.
Change of variables (3D)
dx dy dz = |∂(x,y,z)/∂(u,v,w)| du dv dw
Spherical Jacobian = ρ² sinφ (already inside dV!).
Reading regions
Cone z = √(x²+y²) ⇒ φ = π/4;
sphere x²+y²+z² = a² ⇒ ρ = a.
sphere x²+y²+z² = a² ⇒ ρ = a.
WORKED EXAMPLE
Jacobian of x = r cosθ, y = r sinθ:
det = r cos²θ + r sin²θ = r.
det = r cos²θ + r sin²θ = r.
WATCH OUT!
Mistakes that cost points
- Polar/cylindrical: the r! Spherical: ρ² sinφ!
- φ from the +z-axis (0 → π), θ in the xy-plane
- Jacobian needs the ABSOLUTE value
- Limits must describe the region — sketch first
- Inner limits may depend on outer variables
- φ is from the +z-axis, NOT the xy-plane
- Swapping integration order means NEW limits — re-sketch
- Cylindrical z-limits can depend on r
- Volume = ∫∫∫E 1 dV — write the 1!
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