# Formal Definition of the Definite Integral

## Key Questions

• The definite integral can be used for a number of different purposes.

A common use is finding the area underneath the line on a graph; in this case, the definite integral is taken between the leftmost and rightmost points of the area in question.

Another use - and my personal favourite - is finding the volume, surface area, or similar attributes of solids of revolution. Often, one needs to use the definite integral twice, or three times, in one problem.

• ${\int}_{a}^{b} f \left(x\right) \mathrm{dx} = {\lim}_{n \to \infty} {\sum}_{i = 1}^{n} f \left(a + i \Delta x\right) \Delta x$,
where $\Delta x = \frac{b - a}{n}$