Middle Point Sums
In the Riemann sums, either the left end or the right end of the rectangle touches the curve. In this case, the middle point of the rectangle touches the curve. Everything else is the same as Riemann sums. The figure below shows the function f(x) and different rectangles in the middle point sums.
Let’s say Ai denotes the area of the ith rectangle. The area of this rectangle in this case will be,
Now, the total area in the summation notation will be given by,
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Trapezoidal Rule
The trapezoidal rule is one of the fundamental rules of integration which is used to define the basic definition of integration. It is a widely used rule and the Trapezoidal rule is named so because it gives the area under the curve by dividing the curve into small trapezoids instead of rectangles.
Generally, we find the area under the curve by dividing the area into smaller rectangles and then finding the sum of all the rectangles, but in the trapezoidal rule the area under the curve is divided into trapezoids, and then their sum is calculated. The trapezoidal rule is used to find the value of the definite integrals in numerical analysis. This rule is also called the trapezoid rule or the trapezium rule. Let us learn more about the trapezoidal rule, its formula and proof, example, and others in detail in this article.
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