Definite IntegralsIf you are evaluating a definite integral, the final step is to evaluate it across its limits of integration. This is basically a matter of arithmetic. This work is mainly concerned with more abstract math; mental arithmetic has already been covered well by others, and there are many good books on the subject. One in particular I like and recommend is Math Magic by Scott Flansburg. Nevertheless, there are some techniques and things to consider that are particular to doing abstract math, and which I haven't seen elsewhere. This section discusses some of them.
The best approach depends on the situation. Do you need an exact answer, or will an approximation do? Suppose the correct result is, say, 14.21; if you get 14 as an answer, or even 15, and you know that your answer is close but not exact, will that be good enough for what you need now? The answer really depends on what why you are solving this particular problem. If all you need is an estimate, an approximation will be fine. If you are calculating the final measurements for part of a robotic space satellite, it's important to be as accurate as you can.
If an exact result is necessary, you may be able to calculate it mentally. Again, books such as Math Magic teach how to do this even for complex calculations. Otherwise, you can use a calculator or math software for the final, calculation step.
If an approximation is sufficient, good! You are free to take
shortcuts that will let you get to the solution quickly. You can
round numbers to one or two significant digits (replacing 19.87
with 20, or 1,311 with 1300). You can estimate square roots for
non-square arguments; since 
is
partway between 16 and 25, you might estimate it to be 4.4 or 4.5.
(The actual square root is approximately 4.38.)
If you do decide to do the final number calculation mentally, there are two general strategies that make it easier and faster. They are especially useful when you are approximating.