Hint
The radius at x is x+2. The comparison cylinder has the largest radius of the solid throughout its length.
Worked solution
(a)
Each cross-section is a disk of radius x+2, hence area \(\displaystyle \pi (x+2)^{2}\). Integrate over the x interval from 0 to 2 to obtain the displayed volume.
Answer: \(V=\pi\int_0^2(x+2)^2\,dx\).
(b)
\(\displaystyle V= \pi [\frac{(x+ 2)^{3}}{3}]_{0}^{2}= \frac{\pi (64- 8)}{3}= \frac{56\pi }{3}\) \(\displaystyle \mathrm{cm}^{3}\).
Answer: \(\displaystyle \frac{56\pi }{3}\) \(\displaystyle \mathrm{cm}^{3}\).
(c)
The cylinder volume is \(\displaystyle \pi (4^{2})(2)=32\pi \mathrm{cm}^{3}\). Therefore \(\displaystyle \frac{V}{32\pi }= \frac{\frac{56\pi }{3}}{32\pi }= \frac{7}{12}\). The fraction is dimensionless and less than one.
Answer: \(\displaystyle \frac{7}{12}\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Identify the correct radius x+2.
- 1 mark: Square the radius and include π.
- 1 mark: Use x-limits 0 and 2 with respect to x.
Alternative methods: An expanded squared integrand is accepted.
Common error: Integrating the unsquared radius or using y-limits in an x integral.
(b) · 2 marks
- 1 mark: Use a correct antiderivative of the squared profile.
- 1 mark: Give the exact volume \(\displaystyle \frac{56\pi }{3}\mathrm{cm}^{3}\).
Alternative methods: Expanding \(\displaystyle x^{2}+4x+4\) before integration is accepted.
Common error: Losing π or reporting square units for a volume.
(c) · 2 marks
- 1 mark: Find the comparison-cylinder volume \(\displaystyle 32\pi \mathrm{cm}^{3}\).
- 1 mark: Give the fraction \(\displaystyle \frac{7}{12}\) in the requested order.
Alternative methods: An equivalent simplified exact ratio is accepted.
Common error: Reversing the fraction or using the smaller end radius for the comparison cylinder.
Total: 7 marks.