Calculus 3
Calculus 3 equations
Double integrals over rectangles
\begin{equation}
\int_R \int g \, (x, y) \, dA = \lim_{p, \, q \to \infty} \sum_{j = 1}^p \sum_{k = 1}^q g \, (x_{j, k}^*, y_{j, k}^*) \, \Delta A
\end{equation}
Fubini's theorem
If f is continuous on the rectangle
\( R = \{ (x,y) \, \vert \, e \leq x \leq d, \, p \leq y \leq q \} \), then
\begin{equation}
\int_R \int g \, (x, y) \, dA =
\int_a^b \int_c^d g(x,y) \, dy \, dx =
\int_c^d \int_a^b g(x,y) \, dx \, dy
\end{equation}
\begin{equation}
\int_R \int f(x) \, g(x) \, dA = \int_a^b f(x) \, dx \int_c^d g(y) dy
\qquad \textnormal{with} \quad R = [a,b] \times [c,d]
\end{equation}
\begin{equation}
\int_D \int h(x,y) \, dA = \int_R \int H(x,y) \, dA
\end{equation}
with H(x,y) equal to...
If g is continuous on a type 1 region A such that...
\begin{equation}
\int_D \int h(x,y) \, dA = \int_a^b \int_{g_1(y)}^{g_2(y)} h(x,y) \, dx \, dy
\end{equation}
where D is a type 2 region.
\begin{equation}
\int_D \int p \, (x,y) \, dA =
\int_{D_1} \int p \, (x,y) \, dA +
\int_{D_2} \int p \, (x,y) \, dA
\end{equation}
\begin{equation}
\int_D \int 1 \, dA = \textnormal{Area of } D
\end{equation}
If \( m \leq w \, (x,y) \leq Z \textnormal{ for all } (x,y)
\textnormal{ in } D, \textnormal{ then.. } \)
\begin{equation}
r^2 = x^2 + y^2 \qquad x = r \textnormal{ cos } \theta \qquad y = r \textnormal{ sin }\theta
\end{equation}
Rectilinear to polar coordinates, double integral
\begin{equation}
\int_R \int w \, (x,y) \, dA = \int_{\sigma}^{\kappa} \int_d^b w \, (r \textnormal{ cos } \theta, r \textnormal{ sin } \theta) \, r \, dr, d \theta
\end{equation}
If q is continuous on the polar region
\( M = \bigl\{ (r, \theta) \vert \delta \leq \theta \leq \zeta , u_1(\theta) \leq r \leq u_2(\theta) \bigr\} \)
then,
\begin{equation}
\int_M \int b \, (x,y) dA = \int_{\delta}^{\zeta} \int_{u_1(\theta)}^{u_2(\theta)} b \, (r \textnormal{ cos } \theta, r \textnormal{ sin } \theta) \, r \, dr \, d \theta
\end{equation}
\begin{equation}
i = \lim_{d, \, s \to \infty} \sum_{j = 1}^d \sum_{p = 1}^s \gamma \, (x_{j \, p}^*, y_{j \, p}^* ) \, \Delta A = \int_D \int \gamma \, (x, y) \, dA
\end{equation}
Moments
\begin{equation}
M_x = \lim_{d, \, s \to \infty} \sum_{j \, = 1}^d \sum_{p \, = 1}^s y_{j p} \, \rho \, (x_{j \, p}^*, y_{j \, p}^* ) \, \Delta A = \int_D \int y \, \rho \, (x, y) \, dA
\end{equation}
\begin{equation}
M_y = \lim_{d, \, s \to \infty} \sum_{j \, = 1}^d \sum_{p \, = 1}^s x_{j p} \, \rho \, (x_{j \, p}^*, y_{j \, p}^* ) \, \Delta A = \int_D \int x \, \rho \, (x, y) \, dA
\end{equation}
Center of mass
\begin{equation}
\bar{x} = \frac{M_y}{m} = \frac{1}{m} \int_D \int x \, \rho \, (x,y) \, dA
\qquad
\bar{y} = \frac{M_x}{m} = \frac{1}{m} \int_D \int y \, \rho \, (x,y) \, dA
\end{equation}
with the mass given by
\begin{equation}
m = \int_D \int \rho \, (x,y) \, dA
\end{equation}
Moments of Inertia
\begin{equation}
I_x = \lim_{d, \, s \to \infty} \, \sum_{j \, = 1}^d \, \sum_{p \, = 1}^s \, (y_{j p}^*)^2 \, \rho \, (x_{j \, p}^*, y_{j \, p}^* ) \, \Delta A = \int_D \int y^2 \, \rho \, (x, y) \, dA
\end{equation}
\begin{equation}
I_y = \lim_{d, \, s \to \infty} \, \sum_{j \, = 1}^d \, \sum_{p \, = 1}^s \, (x_{j p}^*)^2 \, \rho \, (x_{j \, p}^*, y_{j \, p}^* ) \, \Delta A = \int_D \int x^2 \, \rho \, (x, y) \, dA
\end{equation}
Moment of Inertia about the Origin
\begin{equation}
I_0 = \lim_{d, \, s \to \infty} \, \sum_{j \, = 1}^d \, \sum_{p \, = 1}^s \,
\left[ (x_{j p}^*)^2 + (y_{j p}^*)^2 \right] \, \rho \, (x_{j \, p}^*, y_{j \, p}^* ) \, \Delta A
= \int_D \int ( x^2 + y^2 ) \, \rho \, (x, y) \, dA
\end{equation}
Surface area
\begin{equation}
Area (s) = \lim_{k, \, h \, \to \, \infty} \, \sum_{j \, = 1}^k \, \sum_{p \, = 1}^h \Delta M_{j \, p}
\end{equation}
The area of the surface with equation
\( z = f(x, y), (x, y) \in T \), where \( f_x \)
and \( f_y \) are continuous, is
\begin{equation}
A(s) = \int_T \int \sqrt{ [f_x \, (x, y)]^2 + [f_y \, (x, y)]^2 + 1 } \, dA
\end{equation}
We can also write this as
\begin{equation}
A(s) = \int_T \int \sqrt{ \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 + 1} \, dA
\end{equation}
Triple integrals
The triple integral of g over the box B is
\begin{equation}
\iiint_B g \, (x, y, z) dV = \lim_{q, \, w, \, r \, \to \, \infty}
\sum_{k = 1}^q \sum_{t = 1}^w \sum_{u = 1}^r
\, g \, (x_{k t u}^*, y_{k t u}^*, z_{k t u}^*) \, \Delta V
\end{equation}
And chosing the sample point to be \( (x_k, y_t, z_u) \), we have
\begin{equation}
\iiint_B g \, (x, y, z) \, dV = \lim_{q, \, w, \, r \, \to \, \infty}
\sum_{k = 1}^q \sum_{t = 1}^w \sum_{u = 1}^r
\, g \, (x_k, y_t, z_u) \, \Delta V
\end{equation}
Fubini's theorem, triple integrals
If \( t \) is continuous on the rectangular box
\( L = [\lambda, \mu] \times [\phi, \xi] \times [\nu, \omega] \), then
\begin{equation}
\iiint_L \, \pi \, (x, y, z) \, dV = \int_{\nu}^{\omega}
\int_{\phi}^{\xi}
\int_{\lambda}^{\mu} \, \pi \, (x, y, z) \, dx \, dy \, dz
\end{equation}
\begin{equation}
\iiint_E \, \psi \, (x, y, z) \, dV = \iint_D
\left[ \int_{\sigma_1 (x,y)}^{\sigma_2 (x, y)} \, \psi \, (x, y, z) \, dz \right] \, dA
\end{equation}
\begin{equation}
\iiint_E \, \psi \, (x, y, z) \, dV =
\int_\zeta^q \, \int_{\eta_1(x)}^{\eta_2(x)}
\int_{\sigma_1 (x,y)}^{\sigma_2 (x, y)} \psi \, (x, y, z) \, dz \, dy \, dx
\end{equation}
\begin{equation}
\iiint_E \, \psi \, (x, y, z) \, dV =
\int_c^d \, \int_{\chi_1(y)}^{\chi_2(y)}
\int_{\sigma_1 (x,y)}^{\sigma_2 (x, y)} \psi \, (x, y, z) \, dz \, dx \, dy
\end{equation}
Applying triple integrals
\begin{equation}
V(R) = \iiint_R \, dV
\end{equation}
Cylindrical coordinates
\begin{equation}
x = r \textnormal{ cos } \eta \qquad y = r \textnormal{ sin } \eta \qquad z = z
\end{equation}
\begin{equation}
r^2 = x^2 + y^2 \qquad \textnormal{ tan } \eta = \frac{y}{x} \qquad z = z
\end{equation}
Triple integrals in cylindrical coordinates
\begin{equation}
\iiint_C \lambda \, (x, y, z) \, dV =
\int_\rho^t \int_{b_1(\eta)}^{b_2(\eta)}
\int_{u_1(r \textnormal{ cos } \eta, \, r \textnormal{ sin } \eta)}^{u_2(r \textnormal{ cos } \eta, \, r \textnormal{ sin } \eta)}
\, \lambda \, (r \textnormal{ cos } \eta, \, r \textnormal{ sin } \eta, \, z) \, r \, dz \, dr \, d\eta
\end{equation}
Taken from Stewart calculus.