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    Calculating The Volume

    Uploaded by

    Ramazan Bayar
    The Volume under the curve

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd

    Calculating The Volume

    Uploaded by

    Ramazan Bayar
    4 views13 pages
    The Volume under the curve

    Original Title

    Calculating the Volume

    Copyright

    © © All Rights Reserved

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    PPTX, PDF, TXT or read online from Scribd

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    The Volume under the curve

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    4 views13 pages

    Calculating The Volume

    Uploaded by

    Ramazan Bayar
    The Volume under the curve

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 13

    CALCULATING THE VOLUME PURE

    OF A SOLID OF MATHEMATIC
    REVOLUTION S
    V=?
    rotated about an
    axis we obtain a
    solid figure. This
    figure is called a
    solid of
    revolution. We can a b
    use the definite
    integral to find the
    volume of a solid of
    revolution.
    EXAMPLE 1:
    What is the volume V of the solid figure generated by rotating the
    area between y = 3x and the x-axis around the x-axis on the
    interval [0, 2]?
    EXAMPLE 2:
    Find the volume of the solid figure generated by rotating the area
    bounded y = 4x – 1, the x-axis, and the lines x = 1 and x = 3
    around the x-axis.
    EXAMPLE 3:
    What is the volume V of the solid obtained by rotating the region
    between y = x^2 + 2 and the x-axis around the x-axis on the
    interval [1, 3]?
    EXAMPLE 4:
    Find the volume of the solid figure generated by rotating the area
    between y = x^2 – 4 and the x-axis around the x-axis.
    NOTE
    If we rotate a figure around
    the y-axis then the volume
    is created by x = f(y) and
    we integrate it with respect
    to dy:
    EXAMPLE 5:
    Find the volume V of the solid figure generated by rotating the region
    between f(x) = 3x – 1, the y-axis, and the lines y = 2 and y = 5
    around the y-axis.
    EXAMPLE 6:
    Find the volume of the solid figure generated by rotating the area
    bounded by y = 2x + 1, the y-axis, and the lines y = 2 and y = 5
    around the y-axis.
    NOTE
    If we rotate the area
    between two curves f(x)
    and g(x) on the interval
    [a, b] then the volume of
    the solid figure
    generated is:
    EXAMPLE 7:
    Find the volume V of the solid figure which is generated by rotating
    the area of the region bounded by the graphs of y = 2x^2 + 2 and
    y = 3 – 2x^2 around the x-axis.
    EXAMPLE 8:
    Find the volume V of the solid figure generated by rotating the area
    bounded by the graphs of y = and y = around the x-axis through
    60°.

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