Today we’re turning away from our work with derivatives to begin studying a new concept,
integrals.
If you'll remember from a previous video in this series, the derivative of a function
at a point is the function's rate of change at that point.
In other words, the derivative equation models where and how fast the original function is
increasing or decreasing.
In contrast, the integral of a function models the area underneath the graph of the function,
and it's calculated opposite of the way we calculated derivatives, which is why an integral
is also called an antiderivative.
Instead of looking for the derivative of a function as we did before, we'll be looking
for the function we would have differentiated to arrive at our current function.
Think about it this way.
From what we know about derivatives, we can see that the derivative of f(x) is g(x), and
that the derivative of g(x) is h(x).
Previously, we were always given g(x) and only asked to calculate the derivative, and
we would have done so and found h(x).
No problem.
With integrals, we'll still be provided with g(x), but this time, we'll be asked to calculate
f(x), it's antiderivative.
Basically we're asking ourselves, what function would I have had to differentiate to get g(x)?
We don't know how to do that yet, but we're going to learn right now.
There are two ways to calculate an antiderivative and find the area underneath the curve: estimate
the area using basic geometric area calculations, or find the exact area using integrals.
If you're taking a standard introductory calculus class, you'll probably learn how to take an
area estimation using Riemann Sums, Trapezoidal Rule, or Simpson's Approximation.
All of these methods are variations on the same theme.
They use rectangles or trapezoids to follow the rough outline of the curve and reduce
the area calculation down to basic geometric formulas.
As you can imagine, the more rectangles or trapezoids you can use, the more accurate
your area estimation can be.
If you imagine that you start using a larger and larger number of rectangles or trapezoids,
until eventually you use an infinite number, this is the point at which you're making an
exact area calculation.
Using this infinite rectangle method to calculate area is similar to using the definition of
the derivative to find rate of change of a function.
Both methods will get you to the right answer, but they're both basic and tedious.
In the same way that we learned better tools for calculating derivatives, like product,
quotient and chain rule, we'll learn better ways to calculate integrals.
Let's start with basic integral notation.
Take this basic polynomial function as an example.
To take it's antiderivative, we'll wrap it inside an integral and a dx.
The integral symbol basically says, take the integral of this function, and the dx says,
with respect to x.
Think about both pieces of notation as a set; they always have to be together when we're
taking antiderivatives.
The integral symbol starts it, and the dx finishes it; they're like bookends that tell
us to take the integral of what comes in between them.
In the same way you're able to take the derivative of a polynomial function by paying attention
to one term at a time, we can take this function's antiderivative by paying attention to one
term at a time.
To simplify, we'll separate each term into its own integral.
When we deal with integrals, we can pull constants out in front of the integral symbol to further
simplify the integration.
Now all we need to do is the opposite of what we've done in the past with derivatives.
Let's focus in on x^3 for a second.
To take the derivative of x^3, we'd bring the 3 down in front, and then subtract 1 from
the exponent, to get a result of 3x^2.
Remember that we want to reverse this process in order to take the integral.
Well, let's do that.
Instead of subtracting 1 from the exponent, we'll add 1.
Then, instead of multiplying the coefficient by the exponent, we'll divide the coefficient
by the exponent.
As you can see, we get an integral of (1/4)x^4.
Did we really do that right?
Well, we can always check ourselves by taking the derivative of our answer.
The derivative of (1/4)x^4 is x^3, our original term, so we know we took the integral correctly.
We can follow this pattern with the rest of the terms in our polynomial function.
We'll always add 1 to the exponent, then divide the coefficient by the new exponent.
Now is a good time to say a quick word about constants.
Remember how constants would disappear when we took their derivatives?
Well, imagine what will happen if we were starting with g'(x) and trying to find g(x),
the antiderivative.
How would we ever know that the "plus 3" was part of the original function?
We wouldn't.
Nor would we know the value of the constant, assuming there was one.
The way we account for this when we integrate is by adding what we call the "constant of
integration" to the antiderivative that we calculate.
We use a generic "+C" to denote it.
Remember that the constant of integration must always be added to your integral function
when you're dealing with indefinite integrals.
Let's return to the integral we were looking at before and give our real final answer by
adding the constant of integration to the end of our integrated function.
Next time we’ll start talking about some techniques we can use to solve more complicated
integrals.
I’ll see you then.