The Trapezium Rule can be used to approximate the value of a definite integral. This is done by dividing up the area into equally wide strips, each of which is considered to be a trapezium. The area under the curve, and thus the definite integral, is given by multiplying the width of the strips with the sum of the average of the first and last vertical and the sum of the rest of the verticals. The more strips used to calculate the area, the more accurate the approximation will be.

The Trapezium Rule only gives an approximation of the area under the curve and not an exact number. If the function is concave up, the approximation will be an overestimate, whereas if the function is concave down, it will be an underestimate.

The area bounded between two curves can be found by subtracting the lower function by the upper one and integrating between the limits.

The Trapezium Rule can be used in modelling a variety of situations.

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The more strips used in the Trapzeium rule, the more accurate the approximation will be.

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The area is given by the integral ∫⁰₋₁x³ + x² + 5 − (x²/2 + 4)dx = ∫⁰₋₁x³ + x²/2 + 1dx = [x⁴/4 + x³/6 + x]⁰₋₁ = 0⁴/4 + 0³/6 + 0 − ((−1)⁴/4 + (−1)³/6 − 1) = 11/12.

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The distance travelled by the marble can be approximated using the Trapezium rule.
Distance = 3(1/2(0.3 + 1.6) + 0.7 + 0.9 + 1.2) = 11.25.

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Diagram A uses more strips, therefore will give a more accurate approximation of the area.

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The distance travelled by the ball can be approximated using the Trapezium rule.
Distance = 4(1/2(0.2 + 1.3) + 0.5 + 0.8 + 1) = 12.2 m.