Multiple Integrals

2.1 Double Integrals

The double integral of $f$ over a rectangle $R = [a,b] \times [c,d]$ is defined as the limit of Riemann sums:

$\iint_R f(x,y)\, dA = \lim_{\lVert P \rVert \to 0} \sum_{i,j} f(x_{ij}^*, y_{ij}^*) \Delta A_{ij}$

Theorem 2.1 (Fubini”s Theorem). If $f$ is continuous on $R = [a,b] \times [c,d]$Then

$\iint_R f(x,y)\, dA = \int_a^b \left(\int_c^d f(x,y)\, dy\right) dx = \int_c^d \left(\int_a^b f(x,y)\, dx\right) dy$

Proof (sketch). For a continuous function $f$ on the compact rectangle $R$Define

$F(x) = \int_c^d f(x,y)\, dy$

Since $f$ is continuous, $F$ is continuous on $[a,b]$. For each partition $P = \\{(x_0, \ldots, x_m)\\}$ of $[a,b]$Define Riemann sums for the outer integral:

$S(P) = \sum_{i=1}^m F(x_i^*)\, \Delta x_i = \sum_{i=1}^m \int_c^d f(x_i^*, y)\, dy\, \Delta x_i$

By Fubini’s theorem for Riemann integrals (proven via uniform continuity of $f$ on the compact set $R$), As $\lVert P \rVert \to 0$ these sums converge to both $\iint_R f\, dA$ and $\int_a^b F(x)\, dx$. The Reversal of integration order follows by symmetry. $\blacksquare$

2.2 General Regions

For a general region $D$ in $\mathbb{R}^2$:

• Type I region: $D = \\{(x,y) : a \leq x \leq b,\, g_1(x) \leq y \leq g_2(x)\\}$

$\iint_D f\, dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\, dy\, dx$

• Type II region: $D = \\{(x,y) : c \leq y \leq d,\, h_1(y) \leq x \leq h_2(y)\\}$

$\iint_D f\, dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y)\, dx\, dy$

Problem. Evaluate $\iint_D xy\, dA$ where $D$ is the region bounded by $y = x^2$ and $y = x + 2$.

Problem. Evaluate $\iint_D x\, dA$ where $D$ is the region bounded by $y = x$, $y = 2x$And $x + y = 2$.

2.3 Triple Integrals

Triple integrals extend to $\mathbb{R}^3$:

$\iiint_E f(x,y,z)\, dV = \iint_D \left(\int_{g_1(x,y)}^{g_2(x,y)} f(x,y,z)\, dz\right) dA$

Problem. Evaluate $\iiint_E z\, dV$ where $E$ is the tetrahedron in the first octant bounded by The coordinate planes and $x + y + z = 1$.

2.4 Change of Variables

Theorem 2.2 (Change of Variables). Let $T : D \subseteq \mathbb{R}^n \to \mathbb{R}^n$ be a $C^1$ diffeomorphism with Jacobian determinant $J_T$. Then

$\int_{T(D)} f(\mathbf{u})\, d\mathbf{u} = \int_D f(T(\mathbf{x}))\, \lvert J_T(\mathbf{x})\rvert\, d\mathbf{x}$

Derivation of the Jacobian factor (for $n = 2$). Let $T(x, y) = (u(x,y),\, v(x,y))$ be a $C^1$ Diffeomorphism. Partition $D$ into small rectangles $R_{ij}$ of area $\Delta x\, \Delta y$. The image $T(R_{ij})$ is approximately a parallelogram spanned by the vectors

$\mathbf{a} = T(x + \Delta x, y) - T(x, y) \approx \left(\frac{\partial u}{\partial x}\Delta x,\, \frac{\partial v}{\partial x}\Delta x\right)$

$\mathbf{b} = T(x, y + \Delta y) - T(x, y) \approx \left(\frac{\partial u}{\partial y}\Delta y,\, \frac{\partial v}{\partial y}\Delta y\right)$

The area of this parallelogram is $\lvert \mathbf{a} \times \mathbf{b} \rvert$Which equals

$\left\lvert \frac{\partial u}{\partial x}\frac{\partial v}{\partial y} - \frac{\partial u}{\partial y}\frac{\partial v}{\partial x} \right\rvert \Delta x\, \Delta y = \lvert J_T \rvert\, \Delta x\, \Delta y$

Summing over all subrectangles and taking the limit gives the change of variables formula. $\blacksquare$

Polar coordinates: $x = r\cos\theta$, $y = r\sin\theta$, $\lvert J \rvert = r$.

$\iint_D f(x,y)\, dA = \iint_{D'} f(r\cos\theta, r\sin\theta)\, r\, dr\, d\theta$

Cylindrical coordinates: $x = r\cos\theta$, $y = r\sin\theta$, $z = z$, $\lvert J \rvert = r$.

$\iiint_E f(x,y,z)\, dV = \iiint_{E'} f(r\cos\theta, r\sin\theta, z)\, r\, dr\, d\theta\, dz$

Spherical coordinates: $x = \rho\sin\phi\cos\theta$, $y = \rho\sin\phi\sin\theta$, $z = \rho\cos\phi$ $\lvert J \rvert = \rho^2 \sin\phi$.

$\iiint_E f(x,y,z)\, dV = \iiint_{E'} f(\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)\, \rho^2 \sin\phi\, d\rho\, d\phi\, d\theta$

2.5 Coordinate System Worked Examples

Problem. Evaluate $\iint_D e^{-(x^2+y^2)}\, dA$ where $D$ is the entire $\mathbb{R}^2$ plane.

Problem. Evaluate $\iiint_E z\, dV$ where $E$ is the solid bounded above by the sphere $x^2 + y^2 + z^2 = 2$ and below by the paraboloid $z = x^2 + y^2$.

Problem. Evaluate $\iiint_E (x^2 + y^2 + z^2)\, dV$ where $E$ is the solid ball $x^2 + y^2 + z^2 \leq a^2$.

2.6 Worked Example

Problem. Compute $\iint_D (x^2 + y^2)\, dA$ where $D$ is the region bounded by $x^2 + y^2 = 4$.

Solution. Use polar coordinates. The region $D'$ is $0 \leq r \leq 2$, $0 \leq \theta \leq 2\pi$.

$\iint_D (x^2 + y^2)\, dA = \int_0^{2\pi} \int_0^2 r^2 \cdot r\, dr\, d\theta = \int_0^{2\pi} \int_0^2 r^3\, dr\, d\theta$

$= \int_0^{2\pi} \left[\frac{r^4}{4}\right]_0^2 d\theta = \int_0^{2\pi} 4\, d\theta = 8\pi$

$\blacksquare$

Problem. Evaluate $\iint_D \frac{y}{x}\, dA$ where $D$ is bounded by $y = x$, $y = 2x$And $x = 1$.

Problem. Swap the order of integration and evaluate: $\int_0^1 \int_{x^2}^1 x e^{y^2}\, dy\, dx$.

2.7 Common Pitfalls

:::caution Common Pitfalls

• Order of integration limits. When setting up $\int_a^b \int_{g_1(x)}^{g_2(x)} f\, dy\, dx$Verify that $g_1(x) \leq g_2(x)$ for all $x \in [a, b]$. If the region is described as “between two curves,” determine which curve is above the other.
• Forgetting the Jacobian. In a change of variables, the Jacobian determinant $\lvert J \rvert$ must be included. For polar coordinates, this factor is $r$; omitting it is one of the most common errors.
• Spherical coordinate conventions. Different texts use different conventions for $\phi$ and $\theta$. Here, $\phi \in [0, \pi]$ is the polar angle (from the positive $z$-axis) and $\theta \in [0, 2\pi]$ is the azimuthal angle.
• Region description. When swapping integration order, carefully redraw the region and re-derive the bounds. The new bounds may require splitting the integral into multiple pieces.

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