An important aspect of using these numerical approximation rules consists of calculating the error in using them for estimating the value of a definite integral. We first need to define absolute error and relative error. Definition: absolute and relative error. In the two previous examples, we were able to compare our estimate of an integral with the actual value of the integral; however, we do not typically have this luxury.
In general, if we are approximating an integral, we are doing so because we cannot compute the exact value of the integral itself easily.
Therefore, it is often helpful to be able to determine an upper bound for the error in an approximation of an integral. The following theorem provides error bounds for the midpoint and trapezoidal rules.
The theorem is stated without proof. Error Bounds for the Midpoint and Trapezoidal Rules. We need to keep in mind that the error estimates provide an upper bound only for the error. The actual estimate may, in fact, be a much better approximation than is indicated by the error bound.
With the midpoint rule, we estimated areas of regions under curves by using rectangles. In a sense, we approximated the curve with piecewise constant functions. One use is when the integrand does not have an antiderivative that is finitely expressible using familiar functions. Many important and interesting functions do not have an antiderivative that can be written using a finite number of simpler functions. An example you may be familiar with is the "bell curve" -- important in probability as relating to the normal distribution.
So we need some kind of approximation method. We can use rectangles, but, in general, trapezoids give us a better approximation with the same number of arithmetic steps.
And there are other ways to approximate. Example 1: Find the area under the curve using trapezoidal rule formula which passes through the following points:. Example 3: Find the area under the curve using the trapezoidal rule formula which passes through the following points:.
Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. The rule is called trapezoidal because when the area under the curve is evaluated, then the total area is divided into small trapezoids instead of rectangles. Then we find the area of these small trapezoids in a definite interval. Learn Practice Download. Trapezoidal Rule In mathematics, the trapezoidal rule, also known as the trapezoid rule or trapezium rule is a technique for approximating the definite integral in numerical analysis.
What is Trapezoidal Rule? Trapezoidal Rule Formula 3. Derivation of Trapezoidal Rule Formula 4. How to Apply Trapezoidal Rule?
Examples Using Trapezoidal Rule Example 1: Find the area under the curve using trapezoidal rule formula which passes through the following points: x 0 0.
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