Approximating Areas Using the Trapezium Rule
The trapezium rule is a numerical method used to estimate the area under a curve. It is particularly useful for irregular shapes where exact integration is difficult. To apply this rule, follow these steps:
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Divide the area into 'n' equal segments.
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Identify the heights of the function at each segment's endpoints.
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Calculate the area of each trapezium using the formula:
[ A = \frac{1}{2} (b_1 + b_2) \times h ]
where ( b_1 ) and ( b_2 ) are the lengths of the parallel sides (heights of the function) and ( h ) is the width of the segment.
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Sum the areas of all trapeziums to get the total area approximation.
For example, to approximate the area under the curve from ( x=1 ) to ( x=3 ) with heights 2 and 4:
- Divide into 1 segment (h = 2).
- Area = ( \frac{1}{2} (2 + 4) \times 2 = 6 ).
Thus, the approximate area is 6 square units.
Key points to remember
- The trapezium rule estimates area under a curve.
- Divide the area into equal segments.
- Use heights at endpoints for calculations.
- Apply the formula for trapezium area.
- Sum areas of trapeziums for total approximation.
Worked example
Calculate the area under the curve from x=0 to x=4 with heights 1, 3, 5, and 2.
- Segments: 3 (h=1).
- Area = (1/2)(1 + 3) + (1/2)(3 + 5) + (1/2)(5 + 2) = 2 + 4 + 3.5 = 9.5.