37 exam-style questions on Trapezoidal rule for approximating areas — topic 5.8 of Calculus in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
The trapezoidal rule with n strips of equal width h estimates the area under a curve. State the correct form of the rule for the integral of y from a to b.
Answer: (h/2)(y0 + yn + 2(y1 + y2 + ... + y(n-1)))
The trapezoidal rule is (h/2)[y0 + yn + 2(y1 + ... + y(n-1))]: the first and last ordinates count once, all interior ordinates count twice.
The trapezoidal rule is used to estimate the area under y = x². For a function that is concave up (curving upward), determine whether the estimate is an overestimate or underestimate of the true area.
Answer: Overestimate
For a concave-up curve each trapezium lies above the curve, so the trapezoidal rule gives an overestimate of the true area.
The speed of a car (m/s) is recorded every 5 seconds: at t = 0, 5, 10, 15 s the speeds are 0, 8, 14, 20 m/s. Using the trapezoidal rule, determine the estimated distance travelled in the 15 seconds.
Answer: 160 m
Distance ≈ area = (5/2)[0 + 20 + 2(8 + 14)] = 2.5 × [20 + 44] = 2.5 × 64 = 160 m.
For which type of function does the trapezoidal rule always give the exact area under the graph?
Answer: Linear functions
A linear function is a straight line, so each trapezium's top coincides exactly with the graph and the rule is always exact; for curved graphs such as quadratics the straight tops generally miss the curve.
Using the trapezoidal rule with 2 strips of equal width to estimate the integral of 2^x from x = 0 to x = 2, determine the estimate.
Answer: 4.5
h = 1; ordinates at x = 0,1,2 are 1, 2, 4. Area = (1/2)[1 + 4 + 2(2)] = (1/2)(9) = 4.5.
Water flows into a tank; the flow rate (litres/min) is measured every minute for 4 minutes: at t = 0,1,2,3,4 the rates are 5, 9, 12, 10, 6. Using the trapezoidal rule, determine the estimated total volume of water added.
Answer: 36.5 litres
h = 1; Volume ≈ (1/2)[5 + 6 + 2(9 + 12 + 10)] = (1/2)[11 + 62] = (1/2)(73) = 36.5 litres.
That was 6 of 37 Trapezoidal rule for approximating areas questions.
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37 Trapezoidal rule for approximating areas questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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