Chapter 32 — Key Takeaways

A compact recap of the chapter's machinery. Every formula below is bookkeeping on one idea: a multiple integral is the limit of a sum of "value times patch-size" over a region.

The Double Integral as Volume and Limit

  • Definition (§32.1): $\displaystyle\iint_R f(x,y)\,dA = \lim_{\|P\|\to 0}\sum_{i,j} f(x_i^*, y_j^*)\,\Delta A_{ij}$ — a double Riemann sum. Each term is the volume of a thin column: floor-patch area $\Delta A$ times height $f$.
  • Geometric meaning: when $f \ge 0$, $\iint_R f\,dA$ is the volume of the solid trapped between the region $R$ and the surface $z = f(x,y)$ — the 2D sibling of "area under a curve" from Chapter 13.
  • Properties (free from the limit): linearity, and additivity over regions — cut an awkward region into friendly pieces and add.

Fubini's Theorem — One Variable at a Time

  • Rectangular case (§32.2): for continuous $f$ on $R = [a,b]\times[c,d]$, $$\iint_R f\,dA = \int_a^b\!\!\int_c^d f\,dy\,dx = \int_c^d\!\!\int_a^b f\,dx\,dy.$$ An iterated integral equals the double integral, and the order does not matter.
  • Procedure: for the inner integral, treat the other variable as a constant; feed the result into the outer integral.
  • Factoring shortcut: $\iint g(x)h(y)\,dA = \big(\int g\,dx\big)\big(\int h\,dy\big)$ — but only when the integrand factors and the region is a rectangle with constant limits.

General Regions and Order of Integration

  • Type I, vertical strips (§32.3): $\displaystyle\int_a^b\!\!\int_{g_1(x)}^{g_2(x)} f\,dy\,dx$ — outer limits constant, inner limits functions of $x$.
  • Type II, horizontal strips: $\displaystyle\int_c^d\!\!\int_{h_1(y)}^{h_2(y)} f\,dx\,dy$.
  • The iron rule: outer limits are always constants; inner limits may depend only on the outer variable.
  • Reversing order: sketch the region, choose the other strip direction, and re-derive fresh limits — never shuffle the old ones in place. Reversal can turn an impossible integral (no elementary inner antiderivative, e.g. $\sin(y^2)$) into an elementary one.

Area, Volume, Average Value

  • Area (§32.4): $\text{Area}(R) = \iint_R 1\,dA$; for Type I this collapses to the area-between-curves formula of Chapter 18.
  • Volume: for $f \ge 0$, $\iint_R f\,dA$ is the volume under the surface.
  • Average value: $\displaystyle\bar f = \frac{1}{\text{Area}(R)}\iint_R f\,dA$.

Polar Coordinates — and the Gaussian Integral

  • Area element (§32.5): $dA = r\,dr\,d\theta$. The factor $r$ is never optional — dropping it is the chapter's single most common error. Sanity check: the unit disk's area is $\pi$, not $2\pi$.
  • Why $r$: a polar patch has sides $dr$ and $r\,d\theta$ (arc length = radius × angle); the factor $r$ is the local area rescaling. This is a preview of the Jacobian determinant, treated in full generality in Chapter 33.
  • The Gaussian payoff: $\displaystyle\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$. Square it, lift to a double integral over $\mathbb{R}^2$, convert to polar; the $r$ in $dA$ makes $u = r^2$ work and $I^2 = \pi$. This normalizes the entire normal distribution (the Chapter 13 anchor) and underlies every $z$-score in statistics.

Triple Integrals and 3D Coordinate Systems

  • Definition (§32.6): $\displaystyle\iiint_E f\,dV = \lim\sum f\,\Delta V$. With $f=1$ it gives volume; with $f = \rho$ it gives mass. The $z$-simple setup integrates $z$ first between two surfaces, then the shadow region as a double integral.
  • Cylindrical (§32.7): $(r,\theta,z)$ with $dV = r\,dr\,d\theta\,dz$ — for an axis of symmetry.
  • Spherical (§32.8): $(\rho,\phi,\theta)$ with $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ — for a point of symmetry. Here $\phi$ is the angle down from the $z$-axis. Yields $V_{\text{ball}} = \tfrac43\pi R^3$ in one product of three integrals.
  • Choosing coordinates (§32.9): match the region's boundary and the integrand — circles/spheres → polar/spherical; flat boxes → Cartesian.

Applications

  • Mass (§32.10): $M = \iint_R \rho\,dA$ or $\iiint_E \rho\,dV$ — mass is the integral of density.
  • Center of mass: $\displaystyle\bar x = \frac{1}{M}\iint x\,\rho\,dA$ (and likewise $\bar y,\bar z$). With constant density this reduces to the geometric centroid.
  • Moment of inertia: $\displaystyle I_z = \iiint_E (x^2+y^2)\,\rho\,dV$ — squared distance from the axis, weighted by mass; measures resistance to spin.
  • Probability (§32.11): a joint density $p(x,y)$ satisfies $\iint_{\mathbb{R}^2} p\,dA = 1$, and $P((X,Y)\in R) = \iint_R p\,dA$. The marginal is $p_X(x) = \int_{-\infty}^\infty p(x,y)\,dy$.

Common Errors to Avoid

  • Dropping the $r$ in polar (or $\rho^2\sin\phi$ in spherical). Always carry the full area/volume element.
  • Shuffling limits when reversing order instead of re-deriving them from the region.
  • Putting a variable in an outer limit — the outermost limits must be pure numbers.
  • Using the factoring shortcut on a non-rectangle or on a sum (e.g. $x + 2y$); it requires a product integrand on a rectangle.
  • Swapping orders on an improper integral without checking absolute integrability (Fubini can fail; see §32.2 Warning).

Connections

  • Backward: generalizes the single-variable definite integral (Chapter 13) and FTC (Chapter 14); reuses area-between-curves (Chapter 18), improper integrals (Chapter 17), and polar coordinates (Chapter 26).
  • Forward: the magic factors $r$ and $\rho^2\sin\phi$ are special cases of the general Jacobian in the change-of-variables formula of Chapter 33 — the multivariable analog of $u$-substitution. Part VII then integrates over vector fields (line and surface integrals, Green/Stokes/Gauss).