Evaluating definite integrals, finding the equation of a curve from its gradient function and a point, and calculating areas bounded by a curve and lines in the first quadrant.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
A definite integral is a number — and that number is an area. This is where integration earns its keep.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Add rectangles and compare with the definite integral.
Copy these into your notebook in your own words — that is how they stick.
∫ₐᵇ f(x) dx = F(b) − F(a).
∫ₐᵇ = −∫ᵇₐ; ∫ₐᵃ = 0.
For f(x) ≥ 0: area under y = f(x) from a to b is ∫ₐᵇ f(x) dx.
y = ∫ (dy/dx) dx + C; use the given point to find C.
The syllabus areas lie in the first quadrant bounded by lines parallel to the y-axis.
The integral is negative; take the modulus for area.
Estimate the area using a rectangle or triangle.
Area in square units.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (25 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [13 marks]
Answer all parts.
(a) Evaluate ∫₁³ (2x + 1) dx. [3]
(b) The gradient of a curve is dy/dx = 3x² − 2 and the curve passes through (1, 4). Find the equation of the curve. [4]
(c) Find the area of the region bounded by y = x² + 2, the x-axis and the lines x = 0 and x = 1. [3]
(d) Find the area of the region in the first quadrant enclosed by y = 4 − x² and the axes. [3]
Question 2 [12 marks]
The curve y = 6x − x² meets the x-axis at O and A.
(a) Find the coordinates of A. [2]
(b) Find the area of the region bounded by the curve and the x-axis. [4]
(c) Find the area under the curve between x = 1 and x = 3. [3]
(d) Evaluate ∫₀^(π/2) (sin x + cos x) dx. [3]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.