Applications Of Double & Triple Integrals

Both double and triple integrals have various applications, especially in physics.

Quantity Functions §

Integrals can be used with any 2D or 3D function that returns a quantity at a given location, $q=q(x,y)$ or $q=q(x,y,z)$.

For an object, $O$, the total quantity is given by:

$$Q_O={\iint }_{A}q(x,y)\text{d}x\text{d}y{Q}_O={\iiint }_{V}q(x,y,z)\text{d}x\text{d}y\text{d}z$$

(The choice depends if we are looking at the object in 2D or 3D)

This quantity can be (but is not limited to):

• Mass
• Charge

Center of Mass §

In a rigidbody, the center of mass is a point where it can be supported such that it does not experience any unbalanced torque that could cause it to rotate. That is, the center of mass is a point where the rigidbody can be supported ‘perfectly’.

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2D Center Of Mass §

For a 2D rigidbody, the location of the center of mass $(x_m,y_m)$ is given by the double integrals:

$$x_m={\iint }_{D}x\cdot \mu (x,y)\text{d}x\text{d}y$$

$$y_m={\iint }_{D}y\cdot \mu (x,y)\text{d}x\text{d}y$$

where $\mu (x,y)$ gives the mass $m$ at a point $(x,y)$

Note, in the trivial case where the mass is uniform at all points ($\mu =1$ everywhere) , we have:

$$x_m={\iint }_{D}x\text{d}x\text{d}y{y}_m={\iint }_{D}y\text{d}x\text{d}y$$

#todo Connection to the expected value $E(X)$ for probability.

3D Center Of Mass §

The 3D center of mass $(x_m,y_m,z_m)$ is analogous to the 2D case, now with an extra dimension:

$$x_m={\iiint }_{D}x\cdot \mu (x,y,z)\text{d}x\text{d}y\text{d}z$$

$$y_m={\iiint }_{D}y\cdot \mu (x,y,z)\text{d}x\text{d}y\text{d}z$$

$$z_m={\iiint }_{D}z\cdot \mu (x,y,z)\text{d}x\text{d}y\text{d}z$$

where $m=\mu (x,y,z)$ returns the mass of the object at a point $(x,y,z)$. In the trivial case it reduces to:

$$x_m={\iiint }_{D}x\text{d}x\text{d}y\text{d}z{y}_m={\iiint }_{D}y\text{d}x\text{d}y\text{d}z{z}_m={\iiint }_{D}z\text{d}x\text{d}y\text{d}z$$

Moments of Inertia §

The moment of inertia of a rigidbody is the resistance of the body when being rotated around a given axis. Like how mass is the resistance of an object to linear acceleration, inertia is an object’s resistance to angular acceleration.

Let $I_n$ be the moment of inertia of a solid body around the $n$-axis. Higher values of $I_n$ imply it is harder to rotated the body around the $n$-axis.

If $\mu$ returns the mass of an object $O$ at a given point $(x,y,z)$, then the moment of inertia is given by:

$$I_x={\iint }_D(y^2+z^2)\cdot \mu (x,y,z)\text{d}x\text{d}y\text{d}z$$

$$I_y={\iint }_D(x^2+z^2)\cdot \mu (x,y,z)\text{d}x\text{d}y\text{d}z$$

$$I_z={\iint }_D(x^2+y^2)\cdot \mu (x,y,z)\text{d}x\text{d}y\text{d}z$$

The reason $I_x$ has a $y^2$ term can roughly be explained:

• #todo