SPM 2021 Add Maths Paper 1, Question 9Integration

Solved on video in English and Bahasa Melayu · Form 5, Integration · 4:50 video

The question

(a)

Diagram 6 shows the shaded region bounded by the curves y = x³ − 6x² + 12x − 6, y = g(x) and the straight line x = 2. The curves intersect at point A(1, 1). It is given that the area under the curve y = g(x) is represented by the definite integral of [−x³ + 4x² − 4x] from x = 1 to x = 2. Find the equation of g(x).

(b)

Find the area of the shaded region bounded by y = x³ − 6x² + 12x − 6, y = g(x) and the line x = 2 (the curves intersect at A(1, 1), and the area under y = g(x) is the integral of [−x³ + 4x² − 4x] from 1 to 2).

SPM 2021 Add Maths Paper 1, Question 9: the question as drawn in Eduly's video4:50

Part (a)

Step 1

The area under y = g(x) is given as the definite integral of [−x³ + 4x² − 4x] from 1 to 2, which means ∫ g(x) dx = −x³ + 4x² − 4x (the antiderivative).

Why this step

The bracket is what you get after integrating g, so it is not g itself. The limits 1 to 2 only give the area, so ignore them here.

2 more steps, each with the reason for it, in the full solution

Final answer

g(x) = −3x² + 8x − 4

Where students lose marks

Watch out: write the result as g(x), not dy/dx. Correct differentiation can still lose the mark if the notation is wrong.

Part (b)

6 more steps, each with the reason for it, in the full solution

Final answer

3/4 unit²

Where students lose marks

Watch out: show the full substitution of the upper and lower limits, or your integration working is marked incomplete and loses marks.

Watch every step of this question solved on video (4:50), with the reason behind each step and a box that checks your own answer.

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