Numerical integration for AQA A-level Maths
This page covers the following topics:
1. Trapezium rule
2. Underestimates and overestimates
3. Calculating area under a curve
4. Numerical integration problems in context
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.
How can using the Trapezium rule to evaluate an integral be made more accurate?
The more strips used in the Trapzeium rule, the more accurate the approximation will be.
Calculate the shaded area of the given graph.
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.
A marble is rolling for 12 seconds. Its speed is recorded every 3 seconds and given in the table. Calculate the distance travelled by the marble.
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.
Which of the two given uses of the Trapezium rule will give a more accurate approximation of the area under the curve?
Diagram A uses more strips, therefore will give a more accurate approximation of the area.
A ball is rolling for 16 seconds. The speed of the ball is recorder ever 4 seconds. Calculate the distance travelled by the ball during its motion.
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.
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