Abstract:
Numerical integration is very important due to its wide usage in many areas such as
Applied Mathematics, Physics, Computational Sciences and Engineering. Finding
precise solution to real-world problems those involving definite triple integrations is
very much complicated and in many occasions it is even impossible. To handle this
situation, computational researchers work with approximation methods. Meantime,
computational scientists work to find or improve existing methods in order to solve
the problem more accurately in short time. In literature, there are numerous digital
integrative approaches discussed but only few papers dealt with approximation of
triple integral. In this paper we suggest an approximation technique for triple
integration that uses second order Taylor polynomial. We consider a function
𝑓(𝑥, 𝑦, 𝑧) of three variables defined on a closed region 𝑅 = [𝑎, 𝑏] × [𝑐, 𝑑] × [𝑒, 𝑓].
Then, we divide the region R into sub-regions as the interval
[𝑎, 𝑏] into l sub-rectangles[𝑥𝑖
, 𝑥𝑖+1] of equal width ∆𝑥 =
𝑏−𝑎
𝑙
, the interval [𝑐, 𝑑]
into m sub-interval [𝑦𝑗
, 𝑦𝑗+1] of equal width ∆𝑦 =
𝑑−𝑐
𝑚
and the interval[𝑒, 𝑓] in to
n sub-intervals [𝑧𝑘, 𝑧𝑘+1] of equal width ∆𝑧 =
𝑒−𝑓
𝑛
. We then use Taylor polynomial
of degree two to find an approximation formula to approximate triple integrals by
considering the given function 𝑓 over each sub-interval by choosing the middle point
of each interval. We applied it to evaluate some selected known algebraic,
trigonometric, exponential, and mixed functions. The results are compared with the midpoint rule and the exact value and observed that generally, we achieved the solution
much quicker and with the least error. Only in the case of a trigonometric function, we got
a higher error for unique dimensions however the accuracy increases with dimensions
