calculus

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Find the volume of the solid formed by rotating the region in the 1st quadrant enclosed by the curves y= x^(1/4) and y= x/64 about the y-axis.

  • calculus -

    If you graph the two functions
    y1(x)=x^(1/4) and
    y2(x)=x/64
    you will see that they meet at two and only two points between which the two curves enclose an area. Rotate this about the y-axis will give the volume.

    The two points are:
    x=0
    y1(0)=0
    y2(0)=0

    The other point can be solved by equating y1(x) and y2(x):
    x^(1/4)=x/64
    cross multiply and switch sides
    x1*x^(-1/4)=64
    x3/4=64
    Take log to solve for x:
    (3/4)log(x)=log(64)
    log(x)=(4/3)log(64)
    Take antilog to get x=4.

    So integration has to be done between 0 and 4.

    The elemental vertical slice is of thickness Δx and height (y1(x)-y2(x)). The area of each slice is therefore (y1(x)-y2(x))Δx.

    Each slice is at a distance x from the y-axis, therefore when revolved around the y-axis it makes a cylinder of volume 2πx * (y1(x)-y2(x))dx

    Integrate this from 0 to 4 will give the required volume.

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