Trapezoidal Rule

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What is the Trapezoidal Rule?

Back

The Trapezoidal Rule is a numerical method used to approximate the definite integral of a function. It works by dividing the area under the curve into trapezoids rather than rectangles, providing a more accurate estimate.

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How do you calculate the area of a trapezoid?

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The area of a trapezoid can be calculated using the formula: Area = (1/2) * (b1 + b2) * h, where b1 and b2 are the lengths of the two parallel sides and h is the height.

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What is the formula for the Trapezoidal Rule?

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The formula for the Trapezoidal Rule is: \( T = \frac{b-a}{2n} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right) \), where [a, b] is the interval, n is the number of trapezoids, and f(x) is the function being integrated.

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If there are 6 ordinates, how many trapeziums are formed?

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If there are 6 ordinates, 5 trapeziums are formed. The number of trapeziums is always one less than the number of ordinates.

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What is the significance of using more trapezoids in the Trapezoidal Rule?

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Using more trapezoids generally increases the accuracy of the approximation of the integral, as it allows for a better fit to the curve of the function.

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What is the error formula for the Trapezoidal Rule?

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The error in the Trapezoidal Rule can be estimated using the formula: \( E_T = -\frac{(b-a)^3}{12n^2} f''(\xi) \), where \( \xi \) is some value in the interval [a, b].

Front

What is the relationship between the number of intervals and the accuracy of the Trapezoidal Rule?

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The accuracy of the Trapezoidal Rule improves as the number of intervals (n) increases, reducing the width of each trapezoid and allowing for a better approximation of the area under the curve.

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Can the Trapezoidal Rule be used for functions that are not continuous?

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The Trapezoidal Rule can be used for functions that are not continuous, but the accuracy of the approximation may be significantly affected.

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What is the first step in applying the Trapezoidal Rule?

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The first step in applying the Trapezoidal Rule is to divide the interval [a, b] into n equal subintervals, determining the width of each subinterval.

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How does the Trapezoidal Rule compare to the Rectangle Method?

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The Trapezoidal Rule generally provides a more accurate approximation than the Rectangle Method because it accounts for the slope of the function by using trapezoids instead of rectangles.