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Calculus II

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Consider the solid obtained by rotating the region bounded by the given curves about the y-axis.
y = ln x, y = 4, y = 5, x = 0
Find the volume V of this solid.

Help!!! Thank you in advance :(

  • Calculus II - ,

    This problem can be easily solved using the disk method.

    Horizontal disks are used, with slices of thickness dy.

    We will integrate from y=4 to y=5.

    Each disk has a volume of πr(y)²dy.
    where the radius is a function of y.

    Since y=ln(x), its inverse relation is x=e^y.

    Integrate for y=4 to 5 of
    V=∫π(e^y)²dy
    =π∫e^(2y)dy
    =π(1/2)e^(2y)
    Evaluate between 4 and 5 gives
    V=(π/2)(e^(2*5)-e^(2*4))
    =29917 (approx.)

    Check:
    The average radius is between e^4 and e^5=101.5
    Volume = 32400 approx. > 29917
    Since the curve ln(x) is concave up, the actual volume should be a little less than the approximation. So the calculated volume should be correct.

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