In the previous post, evaluation of double and triple integrals was discussed. These integrals are sometimes referred to as surface integrals and volume integrals in specific cases; e.g. Gauss’ divergence theorem, Stokes’ theorem etc.
In this post, we will see some applications of multiple integration in physics.
Area and Volume
The area occupied by/ bounded by 2 curves is obtained by substituting the function as a constant function equal to . Thus,
Limits are to be put appropriately.
The volume occupied by a closed surface is obtained by substituting the function as a constant function equal to . Thus,
Again, limits are to be put according to the data given in the problem.
NOTE THAT to get area/volume of a symmetrical shape, things get simplified when the area/volume is evaluated over 1st quadrant/octant and multiplied with appropriate no. of divisions. e.g. volume of an ellipsoid given by the equation
will be equal to times its volume in the first octant.
We know that
So, if is a function of or ,
Mean and Root Mean Square (RMS) Values
The mean value of over region is given by
The mean value of over volume is given by
The RMS values are useful in electrical circuits. For a periodic function with period , the RMS value is given by
Center of Mass, Center of Gravity
The concepts of center of mass and center of gravity have already been discussed in mechanics. So, only mathematical formulation is given.
Consider an elementary mass and density as a function of position of the mass. Now, the C.G. is calculated as
I) If C.G. of a line or an arc is to be computed,
where is the arc length given by
II) If C.G. of a planar laminar is to be computed,
So, the double integral over the lamina needs to be evaluated.
III) If C.G. of a solid object is to be computed,
Moment of Inertia
The moment of inertia, , is a quantity analogous to mass, used in analysis of rotating bodies.
Mathematically, the M.I. of a body about an axis is the second moment of mass about that axis. In other words,
where is the perpendicular distance of the mass from the axis.
The substitutions for will be as explained earlier.
Parallel Axes Theorem states that, if is the M.I. of the body through its centroidal axis, then M.I. of the body about a parallel axis, which is at a distance from the centroidal axis is
where is the mass of the body.
Perpendicular Axes Theorem states that if and are M.I.s of an object about 2 perpendicular axes and respectively, then, M.I. of the object about an axis, which perpendicular to both and is given by