Hint
The partition and endpoint convention are stated. Integrate the continuous function over the interval.
Worked solution
(a)
\([x^3/3+10x^2/2]_0^2=\frac{68}{3}\)
Answer: \(\frac{68}{3}\).
Answer: \(\frac{68}{3}\)
(b)
\(h\sum_{r=1}^n[(rh)^2+a(rh)]=\frac{2665}{108}\)
Answer: \(\frac{2665}{108}\).
Answer: \(\frac{2665}{108}\)
(c)
\(h\sum_{r=0}^{n-1}[(rh)^2+a(rh)]=\frac{2233}{108}\)
Answer: \(\frac{2233}{108}\).
Answer: \(\frac{2233}{108}\)
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: Integrate the continuous function over the interval.
- 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 forget the rectangle width.
(b) · 2 marks
- 1 mark: Include the common width and evaluate the two standard power sums.
- 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: A finite right sum is generally different from the exact area.
(c) · 2 marks
- 1 mark: Start at r=0 and stop at n−1.
- 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 forget the rectangle width.
Total: 6 marks.