Why does integral replace summation?

Why does integral replace summation?

We replace the sum by the integral when the summation is over a continuous region. As, this summation is continuous as we are adding small area elements over a continuous region, so this summation is generally converted to an integral.

When can you interchange summation and integration?

If it is a finite sum, then certainly you can exchange the two. Otherwise, things get more complicated. Let me give a toy example. so if the sum converges absolutely (to an integrable function), then the integral and the summation can be exchanged.

How is integration different from summation?

Summation- Sum of a small numbers of large quantities. Integration- Sum of a large numbers of small quantities. The Summation is a discrete sum whereas Integration is a continuous sum . Here dx is an infinitesimal so that the integral summation is continuous.

Where do we use summation and integration?

The use of Summation arises when there is need of discrete sum of quantities( i.e. addition of few distinct numbers or even large series); Integration is used when the addition is not limited to few or more discrete quantities but the addition is to be done on a continuous basis.

Is integration an infinite sum?

On this leaflet we explain integration as an infinite sum.

Does the order of summation matter?

We also know, however, that when a single series is absolutely convergent, the order of summation does not matter. It turns out that the same is true with double series.

What is the relation between integration and summation?

Integration is basically the area bounded by the curve of the function, the axis and upper and lower limits. This area can be given as the sum of much smaller areas included in the bounded area. Summation involves the discrete values with the upper and lower bounds, whereas the integration involves continuous values.

Is integration Just addition?

The most fundamental meaning of integration is to add up. And when you depict integration on a graph, you can see the adding up process as a summing up of thin rectangular strips of area to arrive at the total area under that curve, as shown in this figure. Look at the thin rectangle in the figure.