EC-18.4 The Integral, and the Two Ways to Read It

What this is and why it exists

Most readers arrive knowing the integral as an antiderivative. That is the reading that matters least in engineering.

The one that matters is the integral as the limit of a sum. Energy delivered over a period of time is that. Charge moved by a varying current is that. The area under a measured curve is that. Each is a total accumulated from pieces, and none of them starts as an antiderivative.

The fundamental theorem is the bridge between the two readings, and it is why the accumulation can be computed at all. This topic spends its time there rather than on a longer list of techniques.

The vocabulary

• Antiderivative — a function whose derivative is the one you started with.
• Indefinite integral — the whole family of antiderivatives, differing by an additive constant.
• Constant of integration — that additive constant, which a physical problem fixes.
• Definite integral — a number, defined as the limit of a sum over an interval.
• Riemann sum — the approximating sum, before the pieces shrink.
• Fundamental theorem of calculus — the result connecting accumulation and differentiation.
• Substitution — the technique that reverses the chain rule.
• Integration by parts — the technique that reverses the product rule.
• Partial fractions — splitting a ratio of polynomials into a sum of simpler ratios.
• Improper integral — one over an unbounded interval, or of a function that grows without bound.

The mental model

Build the definite integral the way it is actually defined, because the definition is the part engineering uses.

Chop an interval into many small pieces. On each piece, multiply the width by a value of the function there. Add the products. Then let the pieces shrink towards nothing. What that sum approaches is the definite integral, and every applied integral in this area is that construction with different quantities substituted in.

The substitutions are worth listing, because seeing the pattern is the skill.

• Width in seconds, value in watts. The sum is energy in joules.
• Width in seconds, value in amperes. The sum is charge in coulombs.
• Width in metres, value in newtons. The sum is work in joules.
• Width in metres, value in a density. The sum is a mass.

In every case the units of the answer are the units of the width multiplied by the units of the value. That is a useful check and it is free.

Now the antiderivative, which is a different question entirely. It asks which function has the one in front of you as its derivative. The answer is never unique, because adding a constant does not change a derivative. That constant is not decoration. In a physical problem it is the initial condition, and dropping it produces an answer that is wrong everywhere rather than wrong at one position.

The fundamental theorem ties the two together. Take the total accumulated up to a position, and treat it as a function of that position. Differentiating it gives back the thing being accumulated. Read the other way, it says a limit of sums can be computed by finding an antiderivative and taking a difference of two values.

That is an extraordinary result and it is worth pausing on. A quantity defined as an infinite limiting process can be obtained by a finite piece of algebra. Nothing else in the subject saves as much labour.

The techniques then follow from the differentiation rules, run backwards.

• Substitution reverses the chain rule. Recognise an inner function and its derivative both present, and change variable.
• Integration by parts reverses the product rule. Use it when a product has one part that becomes simpler when differentiated.
• Partial fractions turns a ratio of polynomials into a sum of pieces, each of which is a standard form. This is the technique that appears again in the transforms course.

Improper integrals close the topic. An integral over an interval with no far end, or of a function that grows without bound near a position, can still come out finite. Deciding which case you are in matters here more than in most calculus courses. Energy over all time is an improper integral, and so is every transform taught later in this area.

The last thing to practise is setting one up. Evaluating an applied integral is rarely the hard part; writing it is. Name a representative thin slice, say what that slice contributes to the total, and choose limits that sweep the slice across the whole region. That procedure is worth repeating until it stops feeling like invention.

What you should now be able to explain or do

• Define the definite integral as a limit of sums and say what quantity it accumulates.
• Explain why the constant of integration is the initial condition of a physical problem.
• State the fundamental theorem in both directions and say why it saves so much labour.
• Apply substitution, integration by parts and partial fractions, and say which differentiation rule each reverses.
• Decide whether an improper integral converges.
• Set up an applied integral from a description by naming the slice and choosing the limits.

Check yourself