Common Misconceptions

⚠️ Common Misconceptions

Misconception 1: "The first and last y-values are multiplied by 2 like the others"

In the trapezium rule, the FIRST (y₀) and LAST (yₙ) y-values appear exactly ONCE in the formula. Only the MIDDLE y-values (y₁ through yₙ₋₁) are multiplied by 2. The formula is: h/2 × (y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ). A common error is multiplying all y-values by 2, which overestimates the area. Remember: "first and last once, all others twice."

Misconception 2: "The trapezium rule gives the exact area under the curve"

The trapezium rule gives an ESTIMATE — an approximation, not the exact area. This is because the tops of the trapezium strips are straight lines, not curves. The actual curve bends between the strip boundaries, so there is always a small gap between the trapezium tops and the actual curve. Increasing the number of strips gives a better estimate, because each strip is narrower and the straight-line tops fit the curve more closely — but it is always an approximation.

Misconception 3: "Area under a speed-time graph is always the distance, even below the x-axis"

On a speed-time graph, speed is always positive (you cannot travel at negative speed), so the graph should always be above the x-axis. However, on a VELOCITY-time graph, negative velocity means travelling in the reverse direction. Area above the x-axis = distance in one direction; area below = distance in the opposite direction. To find TOTAL distance, add the magnitudes (absolute values) of both areas. To find DISPLACEMENT (net change in position), subtract. Always check whether the question asks for distance or displacement.