Engineering Math Practice

Double Integrals

Double integral as a limit of Riemann sums

\iint_R f(x,y)\,dA = \lim_{m,n\to\infty}\sum_{i=1}^{m}\sum_{j=1}^{n} f(x_{ij}^*, y_{ij}^*)\,\Delta A

$\iint_R$ Double integral over the region R — add up contributions across the whole 2D region.
$f(x,y)$ The integrand: the function whose values are being summed.
$dA$ An infinitesimal patch of area — the 2D equivalent of dx.
$\lim_{m,n\to\infty}$ Let the grid get finer in both directions at once: more columns (m) and more rows (n).
$\sum_{i=1}^{m}$ Sum over the m columns of the grid — the x-direction.
$\sum_{j=1}^{n}$ Sum over the n rows of the grid — the y-direction.
$f(x_{ij}^*, y_{ij}^*)$ The function's value at a sample point inside grid cell (i, j).
$\Delta A$ The area of each grid cell — the same for every cell, since the grid is uniform.
$\Delta x$ The width of each grid column.
$\Delta y$ The height of each grid row.

Iterated integral (rectangular region)

\int_c^d\int_a^b f(x,y)\,dx\,dy

$\int_c^d$ Outer integral: sweep y from c to d.
$\int_a^b$ Inner integral: sweep x from a to b, with y momentarily fixed.
$f(x,y)$ The integrand, evaluated at each (x, y) point as the sweep proceeds.
$dx$ Integrate with respect to x first, holding y constant.
$dy$ Then integrate that result with respect to y.

Fubini's theorem (rectangular region)

\iint_{[a,b]\times[c,d]} f(x,y)\,dA = \int_c^d\int_a^b f(x,y)\,dx\,dy = \int_a^b\int_c^d f(x,y)\,dy\,dx

$\iint_{[a,b]\times[c,d]}$ Double integral over the rectangle formed by x in [a,b] and y in [c,d].
$\times$ Cartesian product: pairs every x in [a,b] with every y in [c,d] to form the rectangle.
$f(x,y)$ The same integrand throughout — Fubini's point is that its integral doesn't depend on which variable you integrate first.
$dA$ The area element from the double-integral definition, which the iterated integral reproduces exactly.
$\int_c^d$ Outer integral: sweep y from c to d.
$\int_a^b$ Inner integral: sweep x from a to b, with y momentarily fixed.
$dx$ Integrate with respect to x, holding y constant.
$dy$ Integrate with respect to y, holding x constant.

Double Integrals Over General Regions

Type I region

D = \{(x,y) \mid a \le x \le b,\; g_1(x) \le y \le g_2(x)\}

$a, b$ Fixed numbers — the outer x-bounds, the same for every x in the region.
$g_1(x)$ The lower boundary curve: where the region starts in y, at a given x.
$g_2(x)$ The upper boundary curve: where the region ends in y, at a given x.

Iterated integral over a Type I region

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

$\int_a^b$ Outer integral: sweep x over its fixed interval, exactly like a rectangle's outer bound.
$\int_{g_1(x)}^{g_2(x)}$ Inner integral: for the current x, sweep y from the lower curve up to the upper curve.
$dy\,dx$ Integrate with respect to y first, then with respect to x.

Type II region

D = \{(x,y) \mid c \le y \le d,\; h_1(y) \le x \le h_2(y)\}, \qquad \iint_D f\,dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y)\,dx\,dy

$h_1(y)$ The left boundary curve: where the region starts in x, at a given y.
$h_2(y)$ The right boundary curve: where the region ends in x, at a given y.

Applications of Double Integrals

Mass of a lamina

m = \iint_D \rho(x,y)\, dA

$\rho(x,y)$ Density: mass per unit area, may vary across the plate.
$dA$ An infinitesimal patch of area.
$m$ Total mass of the lamina.

Moments about the x- and y-axis

M_x = \iint_D y\, \rho(x,y)\, dA \qquad M_y = \iint_D x\, \rho(x,y)\, dA

$M_x$ Moment about the x-axis — weights mass by distance from that axis (i.e. by y).
$M_y$ Moment about the y-axis — weights mass by distance from that axis (i.e. by x).

Center of mass

\bar{x} = \dfrac{M_y}{m} \qquad \bar{y} = \dfrac{M_x}{m}

$\bar{x}, \bar{y}$ Coordinates of the center of mass — the balance point of the plate.

Moments of inertia

I_x = \iint_D y^2\, \rho(x,y)\, dA \qquad I_y = \iint_D x^2\, \rho(x,y)\, dA \qquad I_0 = I_x + I_y

$I_x, I_y$ Moments of inertia about the x- and y-axis — weight mass by distance squared; resistance to rotation.
$I_0$ Polar moment of inertia, about the origin — the sum I_x + I_y.

Double Integrals in Polar Coordinates

Polar substitutions

x = r\cos\theta, \qquad y = r\sin\theta, \qquad x^2 + y^2 = r^2

$r$ distance from the origin; the radial coordinate.
$\theta$ angle from the positive x-axis; the angular coordinate.
$x = r\cos\theta,\; y = r\sin\theta$ Cartesian-to-polar conversion.
$x^2 + y^2 = r^2$ the identity that collapses circular integrands.

Area element in polar coordinates

dA = r\, dr\, d\theta

$dA = r\,dr\,d\theta$ the area element; the r is the stretch factor, not optional.

Polar region

D = \{(r,\theta) \;:\; \alpha \le \theta \le \beta,\; h_1(\theta) \le r \le h_2(\theta)\}

$[\alpha,\beta]$ the angular sweep of the region.
$[h_1(\theta), h_2(\theta)]$ inner and outer radial bounds.

Double integral in polar coordinates

\iint_D f(x,y)\, dA = \int_\alpha^\beta \int_{h_1(\theta)}^{h_2(\theta)} f(r\cos\theta,\, r\sin\theta)\; r\, dr\, d\theta

Area in polar coordinates

A(D) = \iint_D 1 \, dA = \int_\alpha^\beta \int_{h_1(\theta)}^{h_2(\theta)} r\, dr\, d\theta

Surface Area

Surface area of z = f(x,y) over D

A(S) = \iint_D \sqrt{1 + \left(f_x\right)^2 + \left(f_y\right)^2}\, dA

$f_x, f_y$ Partial derivatives of f — the surface's slopes; feed the stretch factor.
$dA$ An infinitesimal patch of area in the shadow region D.
$A(S)$ The actual area of the surface S, not just the flat shadow D.

Triple Integrals Over a Box

Triple integral

\iiint_E f(x,y,z)\, dV

$\iiint_E$ triple integral over the solid E.

Rectangular box

B = [a,b] \times [c,d] \times [p,q]

$[a,b],[c,d],[p,q]$ the constant bounds of a rectangular box.

Iterated triple integral over a box

\iiint_B f\, dV = \int_a^b \int_c^d \int_p^q f(x,y,z)\, dz\, dy\, dx

Volume element over a box

dV = dz\,dy\,dx

$dV = dz\,dy\,dx$ a little bit of volume, over a box.

Volume as a triple integral

V(E) = \iiint_E 1\, dV

Triple integral over a general solid

\iiint_E f\, dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f\, dz\, dy\, dx

$g_1(x), g_2(x)$ middle bounds, may depend on x.
$h_1(x,y), h_2(x,y)$ innermost bounds, may depend on x and y.

Triple Integrals in Cylindrical Coordinates

Cylindrical substitutions

x = r\cos\theta, \qquad y = r\sin\theta, \qquad z = z, \qquad x^2 + y^2 = r^2

$r, \theta$ polar coordinates in the xy-plane (distance from the z-axis, angle).
$z$ height, unchanged from Cartesian.
$x^2+y^2=r^2$ collapses circular integrands.

Volume element in cylindrical coordinates

dV = r\, dz\, dr\, d\theta

$dV = r\,dz\,dr\,d\theta$ volume element; the r is the stretch factor, not optional.

Triple integral in cylindrical coordinates

\iiint_E f\, dV = \int_\alpha^\beta \int_{h_1(\theta)}^{h_2(\theta)} \int_{z_1(r,\theta)}^{z_2(r,\theta)} f(r\cos\theta, r\sin\theta, z)\; r\, dz\, dr\, d\theta

$z_1(r,\theta), z_2(r,\theta)$ innermost bounds, may depend on r and θ.

Triple Integrals in Spherical Coordinates

Spherical substitutions

x = \rho\sin\phi\cos\theta, \qquad y = \rho\sin\phi\sin\theta, \qquad z = \rho\cos\phi, \qquad x^2+y^2+z^2 = \rho^2

$\rho$ distance from the origin; ρ ≥ 0.
$\phi$ angle down from the positive z-axis; 0 ≤ φ ≤ π (latitude from the north pole).
$\theta$ angle around the z-axis in the xy-plane; 0 ≤ θ ≤ 2π (longitude).
$x^2+y^2+z^2=\rho^2$ collapses spherical integrands.

Volume element in spherical coordinates

dV = \rho^2\sin\phi\; d\rho\, d\phi\, d\theta

$dV = \rho^2\sin\phi\, d\rho\, d\phi\, d\theta$ the volume element; ρ²sin φ is the Jacobian, never optional.

Triple integral in spherical coordinates

\iiint_E f\, dV = \int_\alpha^\beta \int_{\phi_1}^{\phi_2} \int_{\rho_1}^{\rho_2} f\,\rho^2 \sin\phi\; d\rho\, d\phi\, d\theta

Change of Variables and the Jacobian

The Jacobian (2D)

\dfrac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\[6pt] \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{vmatrix} = \dfrac{\partial x}{\partial u}\dfrac{\partial y}{\partial v} - \dfrac{\partial x}{\partial v}\dfrac{\partial y}{\partial u}

$u, v$ The new coordinates you transform into.
$\dfrac{\partial(x,y)}{\partial(u,v)}$ The Jacobian — determinant of the matrix of partial derivatives.
$|J|$ Absolute value of the Jacobian — the local area-stretch factor.

Change of variables (2D)

\iint_R f(x,y)\, dA = \iint_S f\bigl(x(u,v),\, y(u,v)\bigr)\, \left|\dfrac{\partial(x,y)}{\partial(u,v)}\right| \, du\, dv

$R$ The original region, in the xy-plane.
$S$ The image of R in the uv-plane, under the transformation.

Sanity check: the polar Jacobian

\dfrac{\partial(x,y)}{\partial(r,\theta)} = \begin{vmatrix} \cos\theta & -r\sin\theta \\[6pt] \sin\theta & r\cos\theta \end{vmatrix} = r\cos^2\theta + r\sin^2\theta = r

Multiple Integrals — Reference Sheet