These eight questions are original. They follow the order of the chapter, so early questions use the power rule and later ones combine limits, regions and motion.
Attempt each one with full working before opening the answer.
Questions
Question 1. Find ∫ (8x³ − 6x + 5) dx.
Answer
Integrate term by term: 8x⁴ ÷ 4 − 6x² ÷ 2 + 5x + c = 2x⁴ − 3x² + 5x + c.
Check by differentiating: 8x³ − 6x + 5.
Question 2. A curve has gradient dy/dx = 2x − 6 and passes through (4, 1). Find its equation.
Answer
y = x² − 6x + c. Substituting (4, 1): 16 − 24 + c = 1, so c = 9.
The equation is y = x² − 6x + 9.
Question 3. Evaluate ∫ 3√x dx from 1 to 4.
Answer
3√x = 3x^(1/2), so the integral is 3 × (2/3)x^(3/2) = 2x^(3/2).
[2x^(3/2)] from 1 to 4 = 2(8) − 2(1) = 14.
Question 4. Given ∫ (4x + 1) dx from 0 to k equals 10, and k > 0, find k.
Answer
[2x² + x] from 0 to k = 2k² + k = 10, so 2k² + k − 10 = 0.
Factorise: (2k + 5)(k − 2) = 0, so k = 2 or k = −5/2.
Since k > 0, k = 2. Check: 2(4) + 2 = 10.
Question 5. Find the area enclosed by y = 9 − x² and the x-axis.
Answer
The curve meets the axis where x = −3 and x = 3, and lies above it between them.
Area = [9x − x³/3] from −3 to 3 = (27 − 9) − (−27 + 9) = 18 + 18 = 36 square units.
Question 6. Find the area enclosed by y = x² and y = 4x − x².
Answer
Set x² = 4x − x², so 2x² − 4x = 0 and x = 0 or x = 2.
At x = 1, the second curve gives 3 and the first gives 1, so y = 4x − x² is on top.
Area = ∫ (4x − 2x²) dx from 0 to 2 = [2x² − 2x³/3] from 0 to 2 = 8 − 16/3 = 8/3 square units.
Question 7. The region under y = 2x + 1, between x = 0 and x = 1, is rotated about the x-axis. Find the volume in terms of π.
Answer
V = π ∫ (2x + 1)² dx from 0 to 1 = π ∫ (4x² + 4x + 1) dx.
[4x³/3 + 2x² + x] from 0 to 1 = 4/3 + 2 + 1 = 13/3.
So V = 13π/3 cubic units.
Question 8. A particle has velocity v = 12t − 3t² m/s for 0 ≤ t ≤ 5. Find its displacement and its total distance.
Answer
F(t) = 6t² − t³. Since v = 3t(4 − t), v = 0 at t = 4.
F(0) = 0, F(4) = 96 − 64 = 32 and F(5) = 150 − 125 = 25.
Displacement = 25 − 0 = 25 m.
Distance = |F(4) − F(0)| + |F(5) − F(4)| = 32 + 7 = 39 m.
If you got these wrong
- Questions 1 and 2: read finding an indefinite integral and its constant.
- Questions 3 and 4: study evaluating definite integrals.
- Questions 5 and 6: work through finding area between a curve and an axis and finding area between two curves.
- Question 7: revisit calculating volumes of revolution.
- Question 8: see connecting integration to displacement and distance.
Log each slip in the mistake log and retry after a few days. The timed practice session builder can set up a timed round.
For a teacher to watch your working, see online one-to-one Additional Mathematics tuition, or go back to the integration chapter.