concept definite integral in category python
This is an excerpt from Manning's book Math for Programmers: 3D graphics, machine learning, and simulations with Python MEAP V11.
A definite integral tells you the total change in a function on some interval from its derivative function. The function and a pair of start and end values for the argument, which in our case is time, specify the definite integral. The output is a single number, which gives the cumulative change. For instance, if f(x) is our function of interest and f’(x) is the derivative of f(x), then the change in f from x = a to x = b is f(b) − f(a), and it can be found by taking a definite integral (figure 8.30).
Figure 8.30: The definite integral takes the rate of change (derivative) of a function and a specified interval and recovers the cumulative change in the function on that interval.
Our volume_change function approximates definite integrals, and as we saw in section 8.4.3, it also approximates the area under the flow rate graph. It turns out that the definite integral of a function on an interval is equal to the area under the rate graph on that interval. For most rate functions you meet in the wild, the graphs will be nice enough that you can approximate the area underneath them with skinnier and skinnier rectangles, and your approximations will converge to a single value.
After taking a definite integral, let’s look at an indefinite integral. The indefinite integral takes the derivative of a function and recovers the original function. For instance, if you know that f’(x) is the derivative of f(x), then to reconstruct f(x), you have to find the indefinite integral of f’(x).
The catch is that the derivative f’(x) on its own is not enough to reconstruct the original function f(x). As we saw with get_volume_function, which computed a definite integral, you need to know an initial value of f(x), like f(0) for instance. The value of f(x) can then be found by adding a definite integral to f(0). Because