Uploaded by TheIntegralCALC on 11.06.2012

Transcript:

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.

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.