Hint
The curve is smooth on the stated interval. Square dy/dx inside the arc-length formula.
Worked solution
(a)
\(\int_0^1\sqrt{x+8}\,dx=\frac23[(x+8)^{3/2}]_0^1=2.91505533468698\)
Answer: \(2.91505533468698\).
Answer: \(\approx 2.91505533468698\)
(b)
\(S/\pi=2\int_0^1\frac23(x+7)^{3/2}\sqrt{x+8}\,dx=79.9547212575778\)
Answer: \(79.9547212575778\).
Answer: \(\approx 79.9547212575778\)
(c)
\(V/\pi=\int_0^1\frac49(x+7)^3\,dx=\frac{565}{3}\)
Answer: \(\frac{565}{3}\).
Answer: \(\frac{565}{3}\)
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 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.