Problem
2021 HIHER SCHOOL CERTIFICATE-EXAMINATION PAPER Question 13 (14 marks) Use the Question 13 Writing Booklet (a) A 2-metre-high sculpture is to be made out of concrete. The sculpture is formed 3 by rotating the region between $y=x^{2}, y=x^{2}+1$ and $y=2$ around the $y$-axis. Find the volume of concrete needed to make the sculpture. (b) When an object is projected from a point $h$ metres above the origin with initial 4 speed $V \mathrm{~m} / \mathrm{s}$ at an angle of $\theta^{\circ}$ to the horizontal, its displacement vector, $t$ seconds after projection, is \[ \underset{\sim}{r}(t)=(V t \cos \theta)_{\sim}^{i}+\left(-5 t^{2}+V t \sin \theta+h\right) \underset{\sim}{j} \] (Do NOT prove this.) A person, standing in an empty room which is $3 \mathrm{~m}$ high, throws a ball at the far wall of the room. The ball leaves their hand $1 \mathrm{~m}$ above the floor and $10 \mathrm{~m}$ from the far wall. The initial velocity of the ball is $12 \mathrm{~m} / \mathrm{s}$ at an angle of $30^{\circ}$ to the horizontal. Show that the ball will NOT hit the ceiling of the room but that it will hit the far wall without hitting the floor. Question 13 continues on the following page 346
Solution
Step 1 ：Find the x values for the outer and inner curves at y = 2: \(x_{outer} = \pm\sqrt{2}\), \(x_{inner} = \pm1\)
Step 2 ：Set up the integral to find the volume of the sculpture: \(V = \int_{1}^{2} \pi (r_{outer}^2 - r_{inner}^2) dy\)
Step 3 ：Evaluate the integral: \(V = \pi (2 - 1)\)
Step 4 ：\boxed{V = \pi \text{ cubic meters}}
