One area that I read about a real world example of when approximation of an integral is used is actually in space. Being able to approximate the gravitational force applied by irregular shaped objects, like asteroids that are not spherical in nature, or a planet that is spherical. So with the approximations, if they're able to obtain a lot of data points, they could get better and better approximations through trapezoidal rule or something like Simpson's rule which instead of straight linear lines, the Simpson rule accounts for the curvature of the function between each interval. This is important because understanding the gravitational forces of an object in space can be used for satellite pathing, space travel and space exploration.

As we know from the Riemann Sums learning, the smaller the width (deltax) the better the approximation, this is because you are subdividing the area with more and more rectangles that can more accurately reflect the area. This is why for numerical integration, these approximations become better and better with more data points to use, and so while it won't be the exact area, it's important enough to collect as much data as you can so that when you apply the numerical integration methods (Trapezoidal, Midpoint, Simpson's, etc) you're getting the most accurate answer you can get.

One example that I read about that uses improper integrals is actually in probability and statistics. For instance if you had a continuous random variable X with a PDF function that was and stayed continuous over all reals, what you would be left with is an unbounded, yet continuous function, unbounded in that x could be negative infinity to positive infinity. A function that's unbounded in this way could still provide an area if the function converges at infinity or negative infinity, it is when the function does not converge (diverges) that you will not be able to calculate a number from the definite integral over the unbounded ranges. So for statistics, the expected value calculated with the improper integral represents the average value that the random variable would take if there were an infinite amount of samples or observations. Using this you could have a wide-range of applications including risk analysis and economics.

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