EC-18.7 Multiple Integrals, and Where Calculus Lands in Engineering

What this is and why it exists

This is the last topic of the module and the one that hands the subject on.

A single integral accumulates along a line. Multiple integrals accumulate over an area or a volume, which is what a charge distribution, a stored energy, a mass and a moment all need. Nothing in the electromagnetics course works without them.

It stops deliberately at the integral of a scalar over a region. The integrals of a field, and the theorems relating them, belong to the first electromagnetics topic. That is where they are used, and where the physics motivating them is present.

The vocabulary

• Double integral — an accumulation over a two-dimensional region.
• Triple integral — an accumulation over a three-dimensional region.
• Iterated integral — the same thing computed as one integral inside another.
• Region of integration — the set over which the accumulation runs.
• Order of integration — which variable is swept on the inside and which on the outside.
• Polar coordinates — position given by distance from a centre and an angle.
• Cylindrical coordinates — polar in a plane, with a height added.
• Spherical coordinates — position given by a distance and two angles.
• Jacobian — the factor correcting the size of the area or volume element under a change of variables.
• Centroid — the position of the average of a region, weighted by whatever is distributed over it.

The mental model

The construction is the one from the single-integral topic, applied twice.

Chop a region into many small tiles. On each tile, multiply the tile's area by a value of the function there. Add the products and let the tiles shrink. That limit is the double integral, and a triple integral is the same with small boxes and a volume.

Computing it means turning it into one integral inside another. Sweep along one variable with the other held fixed, then sweep the result. Over a rectangle this is straightforward, because the limits of the inner integral are constants.

Over a general region the difficulty appears, and it is the whole difficulty of the topic. The limits of the inner integral depend on the outer variable. Fix a value of the outer variable. Ask where the region begins and ends along the inner one, and those two answers are the inner limits. They are usually expressions rather than numbers.

This is why the first instruction in any multiple integral is to sketch the region. The sketch is not decoration. It is the only reliable way to read off the limits, and errors in limits are the errors that dominate here.

The sketch also makes the second technique available. The same region can be swept in either order, and one order often gives an integral that can be done while the other does not. Changing the order means re-reading the region the other way round, which is possible only when you have drawn it.

Coordinates come next, and the principle is that the coordinate system should match the symmetry of the region. A circular region is painful in rectangular coordinates and straightforward in polar ones. Anything built around an axis suits cylindrical coordinates. Anything built around a point suits spherical.

Changing coordinates is not free, and the Jacobian is the price. A small rectangle in the new coordinates does not correspond to one of the same size in the old ones. The Jacobian is the factor that corrects for the difference. In polar coordinates that factor is the distance from the centre, which is why the area element carries an extra distance. Omitting it is the most common error in this topic. The answer then comes out wrong by a smooth factor rather than visibly broken.

The applications are then one construction with different quantities substituted in, exactly as in the single-variable case.

• Value equal to a density: the total is a mass or a charge.
• Value equal to one: the total is an area or a volume.
• Value equal to a density multiplied by a position: the total, divided by the mass, is a centroid.
• Value equal to a density multiplied by a squared distance: the total is a moment.
• Value equal to the function itself, divided by the size of the region: the result is an average value.

Recognising that pattern means a new quantity can be set up without looking up a formula for it. That is the point of learning the construction rather than the formulas.

Finally, where this goes. Three courses in this area begin where this module ends. The electromagnetics topic integrates fields rather than scalars, and adds the theorems connecting an integral over a region to one over its boundary. The transforms course integrates a signal against a complex exponential to obtain its frequency content. The differential equations course runs the derivative backwards to recover a signal from its rate of change. All three assume everything in these seven topics and none of them will teach it again.

What you should now be able to explain or do

• Set up a double integral over a general region by sketching it and reading off the limits.
• Change the order of integration and say why it can rescue an intractable integral.
• Convert to polar, cylindrical or spherical coordinates and include the Jacobian.
• Explain what the Jacobian corrects for and what omitting it does to an answer.
• Set up a mass, a centroid, a moment or an average value from the same construction.
• Say which later courses begin where this one stops, and what each of them adds.

Check yourself