Hint
The curve is smooth on the stated interval. Square \(\displaystyle \frac{dy}{dx}\) inside the arc-length formula.
Worked solution
(a)
\(\displaystyle \int_{0}^{3} \sqrt{x+ 10}\, dx= \frac{2}{3}[(x+ 10)^{\frac{3}{2}}]_{0}^{3}= 10.166259986232\)
Answer: \(10.166259986232\).
Answer: \(\approx 10.166259986232\)
(b)
\(\displaystyle \frac{S}{\pi}= 2\int_{0}^{3} \frac{2}{3}(x+ 9)^{\frac{3}{2}}\sqrt{x+ 10}\, dx= 464.522618496548\)
Answer: \(464.522618496548\).
Answer: \(\approx 464.522618496548\)
(c)
\(\displaystyle \frac{V}{\pi}= \int_{0}^{3} \frac{4}{9}(x+ 9)^{3}\, dx= 1575\)
Answer: \(1575\).
Answer: \(1575\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Square \(\displaystyle \frac{dy}{dx}\) inside the arc-length formula.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not replace arc length by a straight-line chord.
(b) · 2 marks
- 1 mark: Multiply radius by the arc element and 2π; evaluate the displayed integral numerically.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Surface area has radius times arc element, not radius squared times dx.
(c) · 2 marks
- 1 mark: Integrate the squared radius for volume.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not replace arc length by a straight-line chord.
Total: 6 marks.