S6 Maths: Integration and Area — Tutor Lesson Plan (60 min, strips to exact, four errors)

2026-11-02

Tutor lesson-prep guides · Secondary mathematics (DSE)

Students integrate correctly and then lose the marks on area, because an integral counts signed area and a shaded region does not. This lesson starts with strips so the integral has a picture before it has a formula.

Objectives — the student can approximate an area with a finite number of strips, evaluate a definite integral using the fundamental theorem, split an interval where the curve crosses the axis, and tell the difference between the value of an integral and the area of a region.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Differentiation	Differentiate x cubed.	3x squared
Indices	What is 3 cubed divided by 3?	9
Function notation	If f(x) = x squared + 1, what is f(2)?	5

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Hook	Shade the region under y = x squared + 1 from 1 to 3	Estimate the area by counting squares
8-20 min	Scene 1	Four strips, then eight, then the limit	Get a lower sum of 8.75
20-30 min	Scene 2	The fundamental theorem as a shortcut	Get 10.67 exactly
30-42 min	Scene 3	Signed area, and the region below the axis	Split the interval at the root
42-52 min	Scene 4	Trapezium rule as an estimate with an error	Compare 10.75 with 10.67
52-60 min	Exit check	Three items in section 08	Marked at once

03 Teaching script (read it out)

Scene 1: Start with strips, not with the integral sign

Cut the interval from 1 to 3 into four strips of width 0.5 and read the height at the left edge of each: 2.00, 3.25, 5.00 and 7.25. Multiply by the width and add: 8.75. That number is too small because the steps sit under the curve. Eight strips give a larger, closer number; the limit of the sum as the number of strips grows is what the integral sign means, not “approximately equal to”.

Ask: “Is 8.75 an overestimate or an underestimate, and why?” The answer depends on whether the curve rises or falls, and saying it out loud is the whole idea of the method.

Scene 2: The fundamental theorem is the shortcut

If F prime is f, then the integral from 1 to 3 of f is F(3) minus F(1). With f(x) = x squared + 1, F(x) = x cubed over 3 plus x. At 3 that is 12, at 1 it is 1.3333, and the difference is 10.6667, which the strips were creeping towards. Teach the notation once: the integral sign is a stretched S for sum, and the numbers at the ends are the ends of the interval.

Warn: the constant of integration cancels in a definite integral. Students who carry a plus c into the subtraction waste a line and sometimes a mark.

Scene 3: Signed area is not area

From minus 1 to 1, the integral of x cubed is zero, because the negative part cancels the positive part, yet the region has an area of 0.5 square units. So an area question needs the roots first, then a split of the interval, then the absolute value of each piece. The trapezium rule with four strips gives 10.75 against an exact 10.6667, an overestimate of 0.0833, because straight tops sit above a curve that bends upwards.

Ask: “Where does the curve cross the axis?” Students who answer that before integrating never subtract a region by accident.

04 Question types and worked steps

Type	Example	Steps	Answer
Evaluation	Evaluate the integral from 1 to 3 of x squared + 1.	Antidifferentiate, substitute both limits, subtract	10.67
Area	Find the area between y = x squared and the x-axis from 0 to 2.	Roots first, then integrate	8/3
Estimate	Estimate the same integral with four strips.	Width times the sum of heights	8.75 for the left sum
Split	Find the area between y = x cubed and the axis from -1 to 1.	Split at the root, take absolute values	0.5

Sanity check (teach this): Every integral answer passes four checks: differentiate the antiderivative back, compare with a rough rectangle, compare with a strip estimate, and check the sign against the picture. A definite integral of a positive function must be positive; a negative value means the limits were substituted in the wrong order, since swapping them flips the sign.

05 Independent practice (3 questions)

Question	Answer	What to watch
Evaluate the integral from 0 to 2 of 3x squared.	x cubed from 0 to 2 = 8.	Positive function, positive answer
Evaluate the integral from 1 to 3 of x squared + 1.	10.67, and the strip estimate 8.75 is below it.	Check the ordering
Find the area between y = x cubed and the axis from -1 to 1.	Split at 0: 0.25 plus 0.25 = 0.5.	Absolute values
Estimate the same area with four strips and say whether it is over or under.	The trapezium rule gives an overestimate because the curve bends upwards.	Direction of the error

06 Common mistakes and what to say

Mistake	Student writes	Say this
Reporting a signed integral as an area	Area is the sum of absolute pieces, an integral is signed	Ask for the roots before the integration
Substituting the limits the wrong way round	Lower limit first, always	Ask which number is on the bottom
Forgetting to divide by the new power	Antidifferentiating x squared gives x cubed over 3	Differentiate the answer back in front of the student
Carrying the constant of integration	It cancels between the two limits	Ask what F(3) minus F(1) does to c

07 Homework (paper, section and question numbers)

HKDSE Mathematics Paper 1 practice paper — Integration questions; redo section 05 questions 2 to 4 of this guide
Secondary mathematics chapter index — Read this chapter with the differentiation chapter

08 Exit check (last 5 minutes)

Question	Answer	Decision
Evaluate the integral from 0 to 1 of 2x.	1	All correct: move to area questions
The integral from -1 to 1 of x cubed is zero. What is the area?	0.5 square units	Zero integral, non-zero area
Is the left sum with four strips above or below the true area for a rising curve?	Below	Rising curve, left edges

09 Tutor FAQ

Do students need to memorise the Riemann sum formula?

They need the picture and one worked sum. The formula is a compression of the picture and is given in the paper when it is needed.

Can a definite integral be negative?

Yes, when the region lies below the axis. That is exactly why an area question has to be split at the roots.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.