Why the expression is worth more than the arithmetic, and the exclusion that catches a keen student.
The expression is the assessed skill. The guidance is explicit that students are expected to write a correct integral expression before calculating an area, and that definite integrals are evaluated with technology. So a lesson spent on by-hand evaluation is spent on the part the calculator does.
That is why the animation prints the expression as prominently as the number, with the limits updating live.
The index is an integer and n is not −1. Adding one to −1 gives zero and you would divide by zero, so 1⁄x is not integrated on this course. A student who has looked ahead or asked an AI will arrive with a logarithm. Say plainly that it is correct mathematics and Higher Level content, so it will not be set for them.
| The area example | ∫₁⁴ (3x² + 2) dx = [x³ + 2x] = (64 + 8) − (1 + 2) = 69. |
| The boundary condition | y = 2x³ + x² + c, and (1, 10) gives 10 = 3 + c so c = 7. |
| 1. The area | 69. |
| 2. Integrate 4x³ − 6x + 5 | B. x⁴ − 3x² + 5x + c. Option C is the same without the constant, deliberately. |
| 3. y when x = 2 | 27, from 16 + 4 + 7. |
1 markThe integral expression with its limits, written down before anything is evaluated. Available even when the rest goes wrong, and routinely thrown away.
1 markThe anti-derivative itself, powers raised and divided correctly.
1 markThe + c on an indefinite integral, or the square brackets with limits on a definite one.
1 markUsing a given point to find c, when there is one.
| They write | What it means |
|---|---|
| 72 (Q1) | Evaluated the top limit and stopped. They have not internalised that a definite integral is a difference. |
| 75 (Q1) | Added instead of subtracting the lower value, or lost a sign inside the bracket. Insist on brackets around the second substitution. |
| Option C (Q2) | The deliberate trap. Everything right except the constant. Mark it wrong and explain that the constant is not decoration. |
| Option A (Q2) | Differentiated. Worth catching early; it usually means the two operations have not separated yet. |
| Option D (Q2) | Raised the powers but did not divide by the new one. The rule was half remembered. |
| 20 (Q3) | Dropped the constant entirely, so the whole point of the boundary condition was lost. |
"Why does the + c disappear in a definite integral?" A good question, and a thirty second answer: it appears at both limits and subtracts away. Show it once rather than asserting it.
"So integration IS area?" Careful. It gives the area while the curve is above the axis. Below it the integral is negative and is no longer the area. At this level questions keep the curve above the axis, but the qualification should be heard now rather than unlearned later.
"Can I just use the calculator?" For the definite integral, yes, and they should. For the anti-derivative and the constant, no.
| What is happening | |
|---|---|
| 1 | Differentiate a few things, then ask what you would start from to GET 6x². Reverse the arrow before naming it. |
| 2 | The rule, with the n is not minus 1 exclusion stated once and clearly. |
| 3 | The + c and the boundary condition. Sketch three parallel curves so the family is visible. |
| 4 | The animation, then the area example. Write the expression line on the board every single time. |
| 5 | Questions 1 to 3. Question 2 option C is the discussion. |
| 6 | Calculator technique for a definite integral, now that they know what it is doing. |
Do not say "integration is the area under the curve" without the qualification. It is true where f is positive, and the unqualified version is something they have to unlearn the first time a curve dips below the axis.
Do not let the + c become a ritual they add without meaning. Spend the two minutes on the family of parallel curves; a student who has seen it stops dropping the constant.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
The lower limit is fixed at 1 and only the upper limit moves, so the expression on screen always matches the shaded region. No external library, nothing stored, nothing sent.