Integrals - Calculus

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.