In calculus, does the signma notation always mean take the antiderivative and then evaluate over the interval?

Answer 1

I think you're slightly confused. ∑ is sigma, and refers to summation. The integration symbol ∫ is called the "long s." And what it refers to is integration. Although the simplest method of evaluating an integral is usually antidifferentiation (especially when exact solutions are desired), this is not the only method, and there are many functions which are integrable, but whose antiderivative cannot be written in terms of elementary functions. e^(x²) is one such function. In these cases, other methods, such as numerical integration or taylor series expansion, must be used to evaluate the integral.

Also, as mentioned by the previous poster, if the ∫ symbol appears without limits it refers to the antiderivative in general (which can still be defined using the integral, even if it cannot be expressed in terms of elementary functions).

Answer 2

To just expand a bit on Pascal's comments:

Unless of course you're referring to Riemann sums, which do use the sigma notation and are a way of simulating an integral without using integration per se. But that's quite different from what you're describing so I think you do indeed mean the long s.

Answer 3

yeah unless there is no interval in which case you just want an antiderivative in general