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Find the volume of the solid formed by rotating the region enclosed by
x=0, \ x=1, \ y=0, \ y= 8 +x^{2}
about the x-axis.

  • Calculus -

    The solid will be rotated about the x-axis, so we can apply the disk method by which vertical slices of thickness dx are integrated.

    Radius of each slice
    = y(x)
    Volume of each slice
    = πr²dx
    = πy(x)²dx

    Total volume can be obtained by integrating from x=0 to 1
    V=∫ πy(x)²dx
    =∫ π (8+x²)² dx
    =∫ π (64+16x+x²)dx
    =π [64x + 8x² + x³/3] x=0 to 1

    Check my work.

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