Maths: Applications and Interpretation · Calculus. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The trapezoidal rule estimates the area under a curve when you cannot integrate exactly, or when you only have a table of measurements. Cut the area into vertical strips of equal width h. Join the top of each strip with a straight line, so each strip becomes a trapezoid, and add the areas of the trapezoids.
In the total, the first and last heights are used once and every height in between is used twice, because each one belongs to two strips.
The answer is an estimate. If the curve bends upwards (concave up) the straight tops lie above the curve, so the estimate is too large. If it bends downwards (concave down) the estimate is too small.
Students double every height, or mix up intervals and heights. Only the middle heights are doubled, and 5 y-values mean 4 intervals, not 5.
Answer: Area ≈ 42 square units, an underestimate (the exact area is 42.7).
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