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  • Integration Link To Riemann Sums

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    msbakermath
    45 views3 pages
    This worksheet is a bit tedius to do, but if the students can have patience, I really do think the repetition they need to endure pays off.

    Original Title

    Integration Link to Riemann Sums

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    This worksheet is a bit tedius to do, but if the students can have patience, I really do think the repetition they need to endure pays off.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOC, PDF, TXT or read online from Scribd
    Flag for inappropriate content
    45 views3 pages

    Integration Link To Riemann Sums

    Uploaded by

    msbakermath
    This worksheet is a bit tedius to do, but if the students can have patience, I really do think the repetition they need to endure pays off.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOC, PDF, TXT or read online from Scribd
    Flag for inappropriate content
    of 3

    Name:__________________

    Below is the graph of f(x) = x


    2
    + 2. The shaded area is the area of concern. We have bounded the area between f(x)
    and the x-axis over the interval [0,1].
    1.)
    a. Let n= 2, so partition the area over the interval [0,1] into 2 equal sections by drawing lines up to
    the f(x) graph. Use these lines as the right-hand endpoints of rectangles.
    b. Determine x . ( Hint x =
    n
    a b
    )
    c. Use the right endpoints to estimate the upper sum of the area under the curve by filling out the
    chart below:
    Height of rectangle
    f(c
    i
    )= f( i x )
    Width of
    rectangle
    x
    Area = hw
    f(c
    i
    ) x
    Rectangle i=1
    2
    2
    2
    +
    ,
    _

    i
    = 2
    2
    1
    2
    +
    ,
    _

    2
    1
    8
    1
    2
    1
    2
    1
    2
    2
    1

    1
    1
    ]
    1

    +
    ,
    _

    Rectangle i=2
    2
    2
    2
    +
    ,
    _

    i
    =
    Total Area:
    d.) The area under the curve is 2.
    33
    or
    3
    1
    2
    . How far off were you with your guess?
    2.)
    a. Let n= 8, so partition the area over the interval [0,1] into 8 equal sections by drawing lines up to
    the f(x) graph. Use these lines as the right-hand endpoints of rectangles.
    b. Determine x . ( Hint x =
    n
    a b
    )
    c. Use the right endpoints to estimate the upper sum of the area under the curve by filling out the
    chart below:
    Height of rectangle
    f(c
    i
    )= f( i x )
    Width of
    rectangle
    x
    Area = hw
    f(c
    i
    ) x
    Rectangle i=1
    2
    8
    2
    +
    ,
    _

    i
    = 2
    8
    1
    2
    +
    ,
    _

    8
    1

    1
    1
    ]
    1

    +
    ,
    _

    8
    1
    2
    8
    1
    2
    Rectangle i=2
    2
    8
    2
    +
    ,
    _

    i
    =
    Rectangle i=3
    Rectangle i=4
    Rectangle i=5
    Rectangle i=6
    Rectangle i=7
    Rectangle i=8
    Total Area:
    d.) Represent the process of summing up the 8 rectangles you just calculated in the chart, using
    sigma/summation notation for representing.
    3.) Draw a picture of the graph f(x) = x
    2
    + 2 over the interval [0,1] such that the area under the curve is
    partitioned into an infinite amount of rectangles bounded by the curve.
    4.) Let n not be defined by a real number. In other words, n is just n, we will partition the interval [0,1] into n
    pieces. Use sigma/summation notation to represent the area under the curve f(x) = x
    2
    + 2 over the interval
    [0,1] if you partition the area into n rectangles.
    5.) Find the limit as n-> of the summation of the n rectangles as you have found in problem number 4. In
    other words find
    x c f
    i
    n
    i
    x


    ) ( lim
    1
    (Hint: Use the summation formulas from page 399 of your textbook to get rid of sigma notation)
    6.) Find F(x) of f(x) = x
    2
    +2. In other words, integrate f(x), or find the antiderivative of f(x).
    Remember the notation: F(x) =

    + 2
    2
    x
    dx.
    7.) Find

    +
    2
    1
    2
    2 x dx by completing parts a c below
    a.) Use your answer from 7 and find F(2)
    b.) Find F(1)
    c.) Use the fundamental Theorem of Calculus to find

    +
    2
    1
    2
    2 x dx = F(2)- F(1)
    8.) Explain the link between limits of Riemann Sums and integration.
    9.) Challenge: Find the area bounded by the equation y = x
    2
    +2 ; y = x + 2 ; x=0 ; and x = 1

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