Equations for Vector Calculus Final (Ch 5.5-5?.?

Polar Double Integrals

\iint f\left(x,y\right)dA=\int_{\alpha}^{\beta}\int_{h_1\left(\theta\right)}^{h_2\left(\theta\right)}f\left(r,\theta\right)rdrd\theta

Cylindrical Triple Integrals (Could be rewritten for any integration order)

\int_{\alpha}^{\beta}\int_{v_1\left(\theta\right)}^{v_2\left(\theta\right)}\int_{u_1\left(r,\theta\right)}^{u_2\left(r,\theta\right)}f\left(r,\theta,z\right)rdzdrd\theta

Spherical Triple Integrals (Could be rewritten for any integration order)

\int_{\psi}^{\delta}\int_{v_1\left(\varphi\right)}^{v_2\left(\varphi\right)}\int_{u_1\left(\theta,\varphi\right)}^{u_2\left(\theta,\varphi\right)}f\left(\rho,\theta,\varphi\right)\rho^2\sin\left(\varphi\right)d\rho d\theta d\varphi

2 Dimensional Total Mass

m=\iint_R\rho\left(x,y\right)dA

3 Dimensional Total Mass

m=\iiint_Q\rho\left(x,y\right)dV

Moment About the X-Axis

M_{x}=\iint_{R}\rho\left(x,y\right)ydA

Moment About the Y-Axis

M_{y}=\iint_{R}\rho\left(x,y\right)xdA

2 Dimensional Center of Mass

\overline{x}=\frac{M_{y}}{m},\overline{y}=\frac{Mx}{m}_{}

2 Dimensional X-Axis Moment of Inertia

I_{x}=\iint_{R}\rho\left(x,y\right)y^2dA

2 Dimensional Y-Axis Moment of Inertia

I_{y}=\iint_{R}\rho\left(x,y\right)x^2dA

2 Dimensional Polar Moment of Inertia

I=I_{x}+I_{y}

Moment About the XY-Plane

M_{xy}=\iiint_{Q}\rho\left(x,y\right)zdV

Moment About the YZ-Plane

M_{yz}=\iiint_{Q}\rho\left(x,y\right)xdV

Moment About the XZ-Plane

M_{xz}=\iiint_{Q}\rho\left(x,y\right)ydV

3 Dimensional Center of Mass

\overline{x}=\frac{M_{yz}}{m},\overline{y}=\frac{M_{xz}}{m},\overline{z}=\frac{M_{xy}}{m}_{}

3 Dimensional X-Axis Moment of Inertia

I_{x}=\iiint_{Q}\left(y^2+z^2\right)\rho\left(x,y\right)dV

3 Dimensional Y-Axis Moment of Inertia

I_{y}=\iiint_{Q}\left(x^2+z^2\right)\rho\left(x,y\right)dV

3 Dimensional Z-Axis Moment of Inertia

I_{z}=\iiint_{Q}\left(x^2+y^2\right)\rho\left(x,y\right)dV

Change in Variable

\left(x,y\right)=T\left(u,v\right)=\left(g\left(u,v\right),h\left(u,v\right)\right)

Jacobian

J = \frac{\partial(x,y)}{\partial(u,v)} = \det \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix}

Change in Variable in Multiple Integration

\iint_{R}f\left(x,y\right)dA=\iint_{S}f\left(u,v\right)\left\vert J\right\vert dudv