HKDSE M2 | Definite Integrals and Areas: tutor prep guide | Riemann sums, trapezium rule and signed area

2026-11-01

Tutor lesson-prep guides · Secondary Mathematics

Students can recite the power rule for integration, but ask what the integral means and they only say “a calculation”. This lesson ties area, limits and signed area into one line so that every question starts with “how many strips?”.

Objectives — Students can: (1) explain a definite integral as the limit of a Riemann sum; (2) evaluate definite integrals with the fundamental theorem and handle the limits correctly; (3) separate signed area from total area and split at the zeros.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Basic differentiation	d/dx of x cubed over 3	x squared
Summation formula	Sum of k squared from 1 to n	n(n+1)(2n+1)/6
Substitution	Order of F(b) and F(a)	upper limit minus lower limit

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	The area problem	No formula covers an arbitrary curve	Say why strips are needed
8-22 min	Riemann sums	Split, sample, add, take the limit	Work n = 4 by hand
22-34 min	Fundamental theorem	F(b) minus F(a) replaces the limit	Obtain 10.667
34-44 min	Trapezium rule	Approximation and error direction	Obtain 10.75
44-54 min	Negative area	Split where the curve crosses the axis	Take absolute values
54-60 min	Exit check	Three questions marked at once	Independent work

03 Teaching script (read it out)

Scene 1: Area: why strips

The area under y = x squared + 1 from x = 1 to x = 3 has no ready formula. Cut the interval into 4 strips of width 0.5, take the left-end heights (2.00, 3.25, 5.00, 7.25), multiply by the width and add: that is the lower sum. More strips means the staircase hugs the curve more closely. With n strips the sum is the sum of (x squared + 1) times delta x, and only the limit as n goes to infinity is the definite integral.

Task: compute the lower sum with 4 strips, then with 8, and compare how much the gap shrinks.

Scene 2: The fundamental theorem: skip the limit

If F prime of x equals f of x, then the integral from 1 to 3 of f is F(3) minus F(1). Take F = x cubed over 3 plus x: (9.0000 + 3) minus (0.3333 + 1) = 10.6667. The trapezium rule with 4 strips gives 10.75, which is 0.0833 too high, because the curve bends upwards and each trapezium roof is a straight chord above it.

Task: explain why an upward-bending curve always makes the trapezium rule overestimate by 0.0833 here.

Scene 3: Signed area: the integral cancels, area does not

The integral of x cubed from -1 to 1 is 0, yet the actual area is two regions added, not zero. A definite integral is signed area; if the question says “area”, find the zeros first, integrate piece by piece and add absolute values. One more standard result: the integral of 1/x from 1 to e equals 1, because ln e is 1.

Note: whenever a question says “area enclosed”, sketch first, find the intersections, then decide how many pieces.

04 Question types and worked steps

Type	Example	Steps	Answer
Calculation	Definite integral	Find F, substitute, subtract	10.667
Approximation	Trapezium rule	h/2 [y0 + 2 sum of middle + yn]	10.75
Concept	Signed area	Split at zeros, take absolute values	Add both pieces
Explanation	Riemann sum limit	Strips, sum, n to infinity	Area is the limit

Sanity check (teach this): Area is always positive; a definite integral can be negative or zero. If the computed area exceeds the bounding rectangle (2 x 10 = 20), the limits were swapped or the absolute value was skipped.

05 Independent practice (3 questions)

Question	Answer	What to watch
Integral of x squared + 1 from 1 to 3	10.667	F = x cubed over 3 plus x
Trapezium rule with n = 4	10.75, error +0.0833	Upward bend overestimates
Integral of x cubed from -1 to 1 and the true area	Integral 0, area 0.5	A quarter on each side

06 Common mistakes and what to say

Mistake	Student writes	Say this
Swapped limits	Reporting the negative result as the area	“Area cannot be negative, so the order is wrong.”
Forgetting to split	Calling the integral of x cubed from -1 to 1 zero an area	“The integral cancels, area does not.”
Wrong strip width	Using n as h	“h is the width of one strip: (b minus a) over n.”
Treating an estimate as exact	Stopping at 10.75	“Say which way the error goes.”

07 Homework (paper, section and question numbers)

Secondary Mathematics chapter index — All chapters
HKDSE M2 practice paper — Integration applications

08 Exit check (last 5 minutes)

Question	Answer	Decision
Integral of x squared + 1 from 1 to 3	10.667	All three right: move to volumes
Trapezium rule n = 4 and error direction	10.75, overestimate	One wrong: redo strips next lesson
An integral of zero but a non-zero area	x cubed on minus one to one	Two wrong: back to act three

09 Tutor FAQ

Must students memorise Riemann sums?

No. HKDSE mainly tests the concept and the trapezium rule; writing the sum form is enough.

Can a definite integral be negative?

Yes, it is signed area below the axis. Questions about area need splitting and absolute values.

Can the trapezium error be reduced?

Yes, with mid-ordinate or Simpson rule, but the exam only asks for over or under.

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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.