Volumes of solids formed by rotating a region about the x-axis, and finding velocity from acceleration and displacement from velocity.
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.
Rotate a flat region about an axis and you sweep out a solid — a cone, a bowl, a vase. Integration finds its volume. Run the same machinery on acceleration and you get velocity and distance.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Rotate a curve about the x-axis in 3D.
Copy these into your notebook in your own words — that is how they stick.
V = π∫ₐᵇ y² dx (about the x-axis; first-quadrant region).
Expand y² fully before integrating.
v = ∫ a dt + C (use initial velocity).
s = ∫ v dt + D (use initial position).
∫ₜ₁ᵗ² v dt.
Split where v = 0 and add absolute values.
SUVAT not valid unless a is constant.
Differentiate your s to recover v and a.
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 (24 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [13 marks]
Answer all parts, leaving answers in terms of π.
(a) The region enclosed by y = x + 2, the x-axis, the y-axis and the line x = 2 is rotated through 360° about the x-axis. Calculate the volume of the solid formed. [5]
(b) Show that the region enclosed by y = x, the x-axis and x = 3, rotated about the x-axis, gives 9π, and verify with the volume of a cone. [4]
(c) Find the volume formed by rotating the region under y = x² + 1 from x = 0 to x = 1 about the x-axis. [4]
Question 2 [11 marks]
A particle moves in a straight line. Its acceleration at time t seconds is a = 6t − 4 m/s². When t = 0 its velocity is 3 m/s and its displacement is 0.
(a) Find expressions for the velocity and displacement at time t. [5]
(b) Show that the particle is never at rest. [2]
(c) Find the distance travelled by the particle in the third second (between t = 2 and t = 3). [3]
(d) Find the acceleration when t = 2. [1]
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.