Topic Review — Further Integration
This review covers all lessons in Further Integration: definite integrals, area under a curve, area between curves, the trapezoidal rule, and applications of integration. Questions are mixed in difficulty.
Review Questions
- Evaluate ∫25 (3x² − 4x + 1) dx.
- Evaluate ∫01 (e2x + 3) dx. Give an exact answer.
- Find the exact area enclosed between the curve y = 4 − x² and the x-axis.
- Use the trapezoidal rule with n = 4 to estimate ∫02 (x³ + 1) dx. Give your answer to 3 decimal places.
- Find the area of the region enclosed between y = 2x − x² and y = 0 (the x-axis).
- Find the area enclosed between y = x and y = x² − 2.
- A particle has velocity v(t) = 6 − 2t m/s. Find the displacement from t = 0 to t = 5. Then find the total distance travelled.
- The rate of fuel consumption of a vehicle is r(t) = 0.5t + 2 litres per hour. Find the total fuel used over 6 hours, and the average rate of consumption.
- Given that ∫04 f(x) dx = 10, ∫02 f(x) dx = 3, find ∫24 f(x) dx. Also find ∫42 f(x) dx.
- Find the average value of f(x) = sin x on [0, π/2]. Determine the value of c in [0, π/2] where f(c) equals this average value.
- The table gives water flow rate (megalitres/day) for a week:
Use the trapezoidal rule to estimate total flow over 7 days.Day 0 1 2 3 4 5 6 7 Rate 12 15 18 14 10 8 9 11 - Find the value of k such that ∫0k (2x + 1) dx = 12.
- A particle starts at position x = 2 m and moves with velocity v(t) = 3t² − 12t + 9 m/s. Find the position at t = 4 s, and the total distance travelled from t = 0 to t = 4.
- A region is bounded above by y = 4 − x² and below by y = 4x − x² − 3. Find the enclosed area, showing all working.
- Evaluate ∫π/6