MATH 101 Lecture 3: Further

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29 Jul 2016
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Here x and y can stand for anything. If x stood for time and y for velocity than the
integral will tell the distance travelled in the first second by a car whose velocity
could be modelled by the equation e^x.
So we know that e^x is essentially a weird shape that cannot be exactly modelled
by simple shapes like rectangles, circles etc.
But in order to find the area under the curve we will use these shapes to give
some estimated/approximated area. That is we will cut this area in small
rectangular slabs and count the area of these rectangular slabs (which would be
length * breadth).
q) Are vertical rectangles the only shape we can use to calculate the area under curves?
A) No, we can use any shape we want. For starters, we can use horizontal rectangles
that dissect the area. I also think we can use circles to fill up the whole area but this
seems to be easiest way of doing it.
If we notice the earlier mentioned symbol of integration, we would realize that the
snake like symbol means sum of all the rectangles, the written function are all the y
values associated with the different heights of the rectangles and the dx are the change
in x values, that is the width of each rectangle.
The discussion of this topic would become more clear when we study the actual method
of computing Reimann sums that is the combined area of all the infinite rectangles.
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