Use the concept of a limit to explain how you could find the exact value for the definite integral value for a section of your graph?
The definite integral value for a section of a graph is the area
under the graph. To compute the area, one method is to add up the
areas of the rectangles that can fit under the graph. By making the
rectangles arbitrarily narrow, creating many of them, you can
better and better approximate the area under the graph.
The limit of this process is the summation of the areas (height
times width, which is delta x) as delta x approaches zero.
The deriviative of a function is the slope of the function. If
you were to know the slope of a function at any point, you could
calculate the value of the function at any arbitrary point by
adding up the delta y's between two x's, again, as the limit of
delta x approaches zero, and by knowing a starting value for x and
y. Conversely, if you know the antideriviative of a function, the
you know a function for which its deriviative is the first
function, the function in question.
This is exactly how integration works. You calculate the
integral, or antideriviative, of a function. That, in itself, is
called an indefinite integral, because you don't know the starting
value, which is why there is always a +C term. To make it into a
definite integral, you evaluate it at both x endpoints of the
region, and subtract the first from the second. In this process,
the +C's cancel out. The integral already contains an implicit dx,
or delta x as delta x approaches zero, so this becomes the area
under the graph.
