This site in Austria allows you to "draw" a graph by hand (though it may take a bit of practice to get exactly the shape you want) and see an indefinite integral displayed at the same time. You can easily check by eye that the height at any point on their blue graph appears to match the 'area so far' up to that point of your red one (with areas below the x-axis giving a negative contribution).
A completely new approach to the area problem was invented in the late 17th century (mainly by Isaac Newton and Gottfried Leibnitz). The idea was to use the fact that when the right end of the interval is moved a little bit, the change in area under the curve is approximately equal to the height of the curve times the distance moved. So the rate of change of area per unit distance is equal to the height. In other words the height function is the derivative of the area function, or conversely the area function is an antiderivative for the height. So an indefinite integral is an antiderivative. But we have lots of results about derivatives, and these can be "read backwards" to tell us about antiderivatives. The various tricks that can be used to figure out antiderivatives this way are often referred to as "techniques of integration".
Once we have a formula for an indefinite integral or 'area-so-far' function,
we can use it to compute any particular definite
integral of the same function, since the area between x=a and x=b is
equal to the difference between the areas (from a fixed starting point)
up to a and up to b.
Area from a to b = (Area from c to b) - (Area from c to a)
This can often be used to get exact results for definite integrals without having to go through the tedious limiting process.
This connection between the integral (or area) and derivative (or slope) problems is known as the Fundamental Theorem of Calculus. It is important not just as a tool for quickly finding some integrals, but also for helping us to recognize the relationship between processes of change and accumulation in many applied fields. This theorem is what enabled the rapid advances in our understanding of the physical world that underlie the industrial revolution. Its discovery is without a doubt one of the most significant events in all of human history!
You might also check our 'raw list' (of links provided without comment) to see if there are any more examples there that we haven't yet included here.
If you have come across any good web-based illustrations of these and
please do let us know and we will add them here.
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