Original subtitles

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The integral function models the area underneath the graph of a function and it's calculated opposite

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of the way we calculated derivatives which is why an integral is also called an anti derivative instead

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of looking for the derivative of a function as we did before.

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We'll be looking for the function we would have differentiated to arrive at our current function.

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Think about it this way from what we know about derivatives we can see that the derivative of f of x

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is x and that the derivative of g of X is HMX.

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Previously we were always given g of X and only asked to calculate the derivative and we would have

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done so and found H of x no problem with integrals will still be provided with g of x.

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But this time we'll be asked to calculate f of x.

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It's antiderivative.

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Basically we're asking ourselves what function what I have had to differentiate to get g of x.

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We don't know how to do that yet but we're going to learn right now.

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There are two ways to calculate an antiderivative and find the area underneath the curve.

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Estimate the area using basic geometric area calculations or find the exact area using integrals.

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If you're taking a standard introductory calculus class you'll probably learn how to take an area estimation

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using Riemann sums.

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Trapezoidal rule where Simpson's approximation all of these methods are variations on the same theme.

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They use rectangles or trapezoids to follow the rough outline of the curve and reduce the area calculation

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down to basic geometric formulas.

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As you can imagine the more rectangles trapezoids you can use the more accurate your area estimation

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can be.

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If you imagine that you start using a larger and larger number of rectangles or trap asteroids until

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eventually use an infinite number.

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This is the point at which you're making an exact calculation using this infinite rectangle method to

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calculate area.

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Is similar to using the definition of the derivative to find the rate of change of a function.

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Both methods will get you the right answer but they're both basic and tedious in the same way that we

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learned better tools to calculate derivatives like product quotient and chain rule.

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We'll learn better ways to calculate integrals.

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Let's start with basic integral notation.

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Take this basic polynomial function as an example to take its antiderivative.

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We'll wrap it inside an integral and ADX the integral symbol basically says take the integral of this

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function and the DX says with respect to x.

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Think about both pieces of notation as a set they always have to be together when we're taking antiderivatives

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the integral symbol starts it and the DX finishes it.

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They're like bookends that tell us to take the integral of what comes in between them in the same way

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that you're able to take the derivative of a polynomial function by paying attention to one term at

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a time.

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We can take this function antiderivative by paying attention to one term at a time.

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To simplify will separate each term into its own integral when we deal with integrals we can put constant

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out in front of the integral symbol to further simplify the integration.

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Now we need to do is the opposite of what we've done in the past with derivatives.

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Let's focus and execute for a second to take the derivative of x cubed.

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We bring the three down in front and then subtract one from the exponent to get a result of 3 x squared.

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Remember that we want to reverse this process in order to take the integral.

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Well let's do that instead of subtracting one from the exponent.

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We'll add one then instead of multiplying the coefficient by the exponent we'll divide the coefficient

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by the exponent.

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As you can see we get an integral of one fourth X to the four.

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Do we really do that right.

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Well we can always check ourselves by taking the derivative or answer the derivative of one fourth X

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to the fourth is x cubed.

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Our original term.

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So we know we took the integral correctly.

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We can follow this pattern with the rest of the terms in our polynomial function will always add 1 to

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the exponent then divide the coefficient by the new exponent.

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Now is a good time to say a quick word about constants.

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Remember how constants would disappear when we took their derivatives.

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Well imagine what will happen if we're starting with G prime of X and trying to find g of X the antiderivative.

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How would we ever know that the plus 3 was part of the original function.

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We wouldn't nor we know the value of the constant assuming there even was one.

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The way that we account for this is by adding the constant of integration to the antiderivative that

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we calculate.

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We use a generic plus C to denote it.

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Remember that the constant of integration must always be added to your integral function.

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When you're dealing with indefinite integrals let's return to the integral we were looking at before

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and give our real final answer.

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But adding the constant integration to the end of our integrated function.