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Definite integrals: reverse power rule | AP Calculus AB | Khan Academy

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    - [Instructor] Let's evaluate
    the definite integral
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    from negative three to five
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    of four dx.
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    What is this going to be equal to?
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    And I encourage you to pause the video
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    and try to figure it out on your own.
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    All right, so in order to evaluate this,
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    we need to remember the
    fundamental theorem of calculus,
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    which connects the notion
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    of a definite integral and antiderivative.
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    So the fundamental theorem
    of calculus tells us
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    that our definite integral
    from a to b of f of x dx
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    is going to be equal to the antiderivative
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    of our function f, which we
    denote with the capital F,
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    evaluated at the upper bound,
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    minus
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    our antiderivative,
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    evaluated at the lower bound.
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    So we just have to do
    that right over here.
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    So this is going to be equal to,
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    well, what is the antiderivative of four?
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    Well, you might immediately say,
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    well, that's just going to be four x.
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    You could even think of it in
    terms of reverse power rule.
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    Four is the same thing
    as four x to the zero.
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    So you increase zero by one.
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    So it's going to be four x to the first,
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    and then you divide by that new exponent.
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    Four x to the first divided by one,
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    well, that's just going to be four x.
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    So the antiderivative is four x.
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    This is, you could say,
    our capital F of x,
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    and we're going to evaluate that
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    at five and at negative three.
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    We're gonna find the
    difference between these two.
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    So what we have right over here,
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    evaluating the antiderivative
    at our upper bound,
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    that is going to be four times five.
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    And then from that,
    we're going to subtract,
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    evaluating our antiderivative
    at the lower bound.
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    So that's four times negative three.
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    Four times negative three.
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    And what is that going to be equal to?
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    So this is
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    20 and then minus negative 12.
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    So this is going to be plus 12,
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    which is going to be equal to 32.
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    Let's do another example
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    where we're going to do
    the reverse power rule.
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    So let's say
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    that we want to find
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    the indefinite or we want to
    find the definite integral
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    going from negative one
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    to three
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    of seven
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    x squared
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    dx.
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    What is this going to be equal to?
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    Well, what we want to do is
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    evaluate what is the
    antiderivative of this?
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    Or you could say what is,
    if this is lowercase f of x,
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    what is capital F of x?
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    Well, the reverse power rule,
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    we increase this exponent by one.
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    So we're going to have
    seven times x to the third,
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    and then we divide by
    that increased exponent.
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    So seven x to the third divided by three,
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    and we want to evaluate
    that at our upper bound
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    and then subtract from that
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    and it evaluate it at our lower bound.
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    So this is going to be equal to,
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    so evaluating it at our upper bound,
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    it's going to be seven
    times three to the third.
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    I'll just write that three
    to the third over three.
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    And then from that, we
    are going to subtract
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    this capital F of x,
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    the antiderivative evaluated
    at the lower bound.
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    So that is going to be
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    seven times negative one to the third,
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    all of that over three.
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    And so this first expression, let's see,
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    this is going to be seven
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    times three to the third over three.
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    This is 27
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    over three.
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    This is going to be the same
    thing as seven times nine.
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    So this is going to be 63.
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    And this over here,
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    negative one to the third
    power is negative one.
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    But then we were subtracting a negative,
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    so this is just gonna be adding.
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    And so this is just going
    to be plus seven over three.
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    Plus seven
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    over three,
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    if we wanted to express
    this as a mixed number,
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    seven over three is the
    same thing as 2 1/3.
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    So when we add everything together,
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    we are going to get 65 1/3.
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    And we are done.
Title:
Definite integrals: reverse power rule | AP Calculus AB | Khan Academy
Description:

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Video Language:
English
Team:
Khan Academy
Duration:
04:14

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