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Cross-references: intervals, number, Riemann integrable, subinterval, examples of estimating a Riemann integral, trapezoidal rule, line segments, right triangles, right trapezoids, graph, areas, expressions, partition, interval of integration, points, definite integral
There are 4 references to this entry.
This is version 8 of composite trapezoidal rule, born on 2006-07-19, modified 2008-03-12.
Object id is 8147, canonical name is CompositeTrapezoidalRule.
Accessed 4760 times total.
Classification:
| AMS MSC: | 41A05 (Approximations and expansions :: Interpolation) | | | 41A55 (Approximations and expansions :: Approximate quadratures) |
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Pending Errata and Addenda
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![$ [a,b]$](http://images.planetmath.org:8080/cache/objects/8147/l2h/img2.png "$ [a,b]$"){kind=small}
![$\displaystyle \int\limits_a^b f(x)\,dx\; \approx\;\frac{1}{2}\sum_{j=1}^n(x_j-x_{j-1})\left[f(x_{j-1})+f(x_j)\right]. $](http://images.planetmath.org:8080/cache/objects/8147/l2h/img4.png "$\displaystyle \int\limits_a^b f(x)\,dx\; \approx\;\frac{1}{2}\sum_{j=1}^n(x_j-x_{j-1})\left[f(x_{j-1})+f(x_j)\right]. $")
![$\displaystyle \int\limits_a^b f(x)\,dx\; \approx\;\frac{h}{2}\left[ f(a) + 2\sum_{j=1}^{n-1}f(a+jh)+f(b)\right]. $](http://images.planetmath.org:8080/cache/objects/8147/l2h/img7.png "$\displaystyle \int\limits_a^b f(x)\,dx\; \approx\;\frac{h}{2}\left[ f(a) + 2\sum_{j=1}^{n-1}f(a+jh)+f(b)\right]. $")
![$\displaystyle \;\frac{1}{2}\sum^n_{j=1}(x_j-x_{j-1})\left[f(x_{j-1})+f(x_j)\right].$](http://images.planetmath.org:8080/cache/objects/8147/l2h/img13.png "$\displaystyle \;\frac{1}{2}\sum^n_{j=1}(x_j-x_{j-1})\left[f(x_{j-1})+f(x_j)\right].$")
![$ [a,b]$](http://images.planetmath.org:8080/cache/objects/8147/l2h/img15.png "$ [a,b]$"){kind=small}
![$ x \in [a,b]$](http://images.planetmath.org:8080/cache/objects/8147/l2h/img17.png "$ x \in [a,b]$"){kind=small}
![$\displaystyle \left\vert \int\limits_a^b f(x) \, dx - \frac{1}{2}\sum_{j=1}^n(x_j-x_{j-1})\left[f(x_{j-1})+f(x_j)\right] \right\vert \le \frac{M(b-a)^3}{12n^2}. $](http://images.planetmath.org:8080/cache/objects/8147/l2h/img20.png "$\displaystyle \left\vert \int\limits_a^b f(x) \, dx - \frac{1}{2}\sum_{j=1}^n(x_j-x_{j-1})\left[f(x_{j-1})+f(x_j)\right] \right\vert \le \frac{M(b-a)^3}{12n^2}. $")