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SKILL BUILDER: Intuitive Calculus
curve. It is also proportional to the area under
the blue derivative curve, since each Lego brick
contributes one rectangle of area we can add up
to get the whole.
In calculus, adding up a quantity represented
by a curve is called taking its integral. Generally
speaking, an integral is what you get when you
add up what is underneath a curve: the area
under a curve, or the volume under a surface.
Let’s recap what we did.
• We started with a red wall that was 0 bricks
high, then 1, 4, 9, 16, up to 25.
• Then we created a new blue wall made up of
the differences from column to column in the
original wall. This new wall, we discovered,
could be called the derivative of the first one.
• Next, we added up the number of bricks in our
derivative wall. We discovered that the running
total (integral) of the derivative is just the
number of bricks in the original curve at that
point!
To put it another way, finding derivatives and
integrals are what mathematicians would
call inverses of each other, or operations that
cancel each other out. This is similar to the
relationship between multiplication and division,
or squaring and taking a square root. The fact
that the derivative of an integral gives you
back the original curve you started with, and
likewise the integral of the derivative, is called
the Fundamental Theorem of Calculus. And no
equations required (so far)! In more complicated
cases, the integral may have a constant offset, but
we constructed our examples to avoid that.
Continuous-Curve Model
As we just saw, the brick model works for curves
that can be easily chopped up into a small
number of whole bricks. But what if we had a
smooth curve, that was changing in big swoops
in some places and barely changing in others?
Isaac Newton’s big “aha” moment (one of them,
anyway) was that you could create ways of figuring
out the slope of a curve instantaneously, between
two “columns” that are incredibly close together.
We could imagine that we create a zillion columns
of teeny bricks and do the same exercise.
Rich Cameron
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