Riemann Sums
The Riemann sums up work on the idea of diving the area under the curve into different rectangular parts. As the number of rectangles increases, the area becomes closer and closer to the current area. In the figure shown below, there is a function f(x). The area under this function is divided into many rectangles. The total area under the curve is the sum of the areas of all the rectangles.
Notice that in the above figure, the right end of the rectangles touches the curve. This is called right-Riemann sums.
In another case, when the left end of the rectangles touches the curve as shown in the image below, they are called left Riemann sums.
Let’s say Δx is the width of the interval width n is the number of intervals as stated above. Then the area of the curve represented by the sum is given by,
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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