Here we will show the definition of a definite integral (\( \htmlClass{sdt-0000000060}{\int} \))
| \( x \) | This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( i \) | This is the symbol for an iterator, a variable that changes value to refer to a sequence of elements. |
| \( b \) | This is a symbol for any secondary generic constant. It can hold any numerical value |
| \( \int \) | This is the symbol for an integral, sometimes referred to as an antiderivative. Graphically, it can be understood as the area between a curve and the axis the integral is taken with respect to. |
| \( \sum \) | This is the summation symbol in mathematics, it represents the sum of a sequence of numbers. |
| \( f \) | This is the symbol for a function. It is commonly used in algebra, and (multivariate) calculus. |
| \( \lim \) | This is the symbol for a limit in calculus. It's a function that models an output, as an input approaches a certain value. |
| \( \:d \) | This is the symbol for a differential. It represents an infinitesimally small (infinitely close to zero) change in whatever variable it is with respect to. |
| \( n \) | This is the symbol for the number of subintervals an area is broken up into. |
| \( \Delta \) | This is the symbol for the amount that some variable changes. |
| \( \infty \) | This is the symbol for infinity, a concept representing the idea of something without bound or end. It represents an unbounded quantity larger than any real number. |
| \( a \) | This is a symbol for any generic constant. It can hold any numerical value |
Consider that we are trying to calculate an area under a curve with a definite integral:
The symbol \(\int\) represents an integral, sometimes referred to as an antiderivative. Graphically, it can be understood as the area between a curve and the axis the integral is taken with respect to. It is the opposite of a derivative, meaning that \(\int \htmlClass{sdt-0000000065}{f'(x)} \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x}= \htmlClass{sdt-0000000120}{f(x)}\)
In the case of \(\htmlClass{sdt-0000000060}{\int}_{\htmlClass{sdt-0000000121}{a}}^{\htmlClass{sdt-0000000033}{b}}\) we are trying to find the area under the curve between \(\htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000121}{a}\) and \(\htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000033}{b}\), as can be seen by the highlighted purple in the following figure:
Calculating the area directly would be challenging. However, we can much more easily consider calculating an approximation of the area, by splitting the highlighted area into smaller rectangles, as is illustrated in the following figure:
An approximation for the total area will be the sum of the areas of each rectangle.
We will say that the number of intervals (rectangles) is noted by the symbol: \( \htmlClass{sdt-0000000104}{n} \).
Any arbitrary one of these intervals will be noted by the symbol: \(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}\)
We can now use the equation for the area of a rectangle:
\[\htmlClass{sdt-0000000115}{A} = \htmlClass{sdt-0000000107}{W} \cdot \htmlClass{sdt-0000000110}{H}\]
To get the area of each rectangle.
The width of each rectangle will be the change in \(\htmlClass{sdt-0000000003}{x}\) for that interval, \(\htmlClass{sdt-0000000105}{\Delta} \htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}\)
The height of each rectangle will be the value of \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}})\). The area of any given rectangle is therefore:
\[ \htmlClass{sdt-0000000115}{A} = \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) \htmlClass{sdt-0000000105}{\Delta} \htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}} \]
We can now sum these all, to get an approximation of the total area, yielding the equation.
\[ \htmlClass{sdt-0000000115}{A} = \htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000018}{i} = 1}^{\htmlClass{sdt-0000000104}{n}} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) \htmlClass{sdt-0000000105}{\Delta} \htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}} \]
We would now like to go from an approximation of the overall area, to an exact answer. To do this, we can imagine what would happen as we split the area up into more and more rectangles. If we imagine an infinite number of rectangles, each would be infinitely thin, therefore not having any error. This infinitely thin length is a differential, expressed as \(\htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x}\) We can therefore get the exact answer by taking the limit as the number of intervals (\( \htmlClass{sdt-0000000104}{n} \)) approaches infinity (\( \htmlClass{sdt-0000000108}{\infty} \)). Mathematially, this is expressed as:
\[ \htmlClass{sdt-0000000060}{\int}_{\htmlClass{sdt-0000000121}{a}}^{\htmlClass{sdt-0000000033}{b}} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000101}{\lim}_{\htmlClass{sdt-0000000104}{n} \to \htmlClass{sdt-0000000108}{\infty}} \htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000018}{i} = 1}^{\htmlClass{sdt-0000000104}{n}} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) \htmlClass{sdt-0000000105}{\Delta} \htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}} \]
as required.
Coming soon...