Why is Trapezoidal so called
since we can approximates the given integral by the sum of n Trapezoids
both (1) and (2)
None of the above
since we can exact the given integral by the sum of n Trapezoids
Answers
Answer:
In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule—see Trapezoid for more information on terminology) is a technique for approximating the definite integral.
The trapezoidal rule works by approximating the region under the graph of the function {\displaystyle f(x)}f(x) as a trapezoid and calculating its area. It follows that
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite") trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule". Let {\displaystyle \{x_{k}\}}{\displaystyle \{x_{k}\}} be a partition of {\displaystyle [a,b]}[a,b] such that {\displaystyle a=x_{0}<x_{1}<\cdots <x_{N-1}<x_{N}=b}{\displaystyle a=x_{0}<x_{1}<\cdots <x_{N-1}<x_{N}=b} and {\displaystyle \Delta x_{k}}{\displaystyle \Delta x_{k}} be the length of the {\displaystyle k}k-th subinterval (that is, {\displaystyle \Delta x_{k}=x_{k}-x_{k-1}}{\displaystyle \Delta x_{k}=x_{k}-x_{k-1}}), then
{\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}={\tfrac {\Delta x}{2}}\left(f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+2f(x_{4})+\cdots +2f(x_{N-1})+f(x_{N})\right)}{\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}={\tfrac {\Delta x}{2}}\left(f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+2f(x_{4})+\cdots +2f(x_{N-1})+f(x_{N})\right)}.
An animation that shows what the trapezoidal rule is and how the error in approximation decreases as the step size decreases
Illustration of "chained trapezoidal rule" used on an irregularly-spaced partition of {\displaystyle [a,b]}[a,b].
The approximation becomes more accurate as the resolution of the partition increases (that is, for larger {\displaystyle N}N, {\displaystyle \Delta x_{k}}{\displaystyle \Delta x_{k}} decreases). When the partition has a regular spacing, as is often the case, the formula can be simplified for calculation efficiency.
As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule.