MRSM Trial 2025 Add Maths Paper 2, Question 10Integration

Solved on video in English · Form 5, Integration · 8:32 video

The question

(a)(i)

Diagram 7.1 shows a curve x = 9 − y² and a straight line y = 2.

(a)(i) Calculate the area of the shaded region.

(a)(ii)

(a)(ii) Find the volume generated, in terms of π, when the region bounded by the curve x = 9 − y², the straight line y = 1 and the y-axis is revolved through 60° about the y-axis.

(b)

Diagram 7.2 shows part of the curve y = f(x), a symmetric U-shape from x = −b to x = b reaching height b. Given that ∫₀ᵇ x dy = −π, find the area of the shaded region.

MRSM Trial 2025 Add Maths Paper 2, Question 10: the question as drawn in Eduly's video8:32

Part (a)(i)

Step 1

Integrate the curve with respect to y (it meets the y-axis where x = 0, i.e. y² = 9, y = 3): A₁ = ∫₀³ (9 − y²) dy = [9y − y³/3]₀³ = (27 − 9) − 0 = 18.

Why this step

The curve gives x in terms of y, so we add up horizontal strips and integrate with respect to y. Using dx is the classic slip.

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

Final answer

8 unit²

Where students lose marks

Watch out: integrate with respect to y (not x), then remember to subtract the rectangle below y = 2.

Part (a)(ii)

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

Final answer

136π/15

Where students lose marks

Watch out: multiply by 60/360 = 1/6 for a partial turn, use limits y = 1 to 3, and expand (9 − y²)² fully.

Part (b)

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

Final answer

2b² − 2π

Where students lose marks

Watch out: double one corner for the two symmetric strips, and use the size π of the given −π as an area.

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