# The region under the curves #x=0, x=y-y^4# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?

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To sketch the region and find the volumes of the two solids of revolution:

a) For rotation about the x-axis, sketch the region bounded by the curves x = 0 and x = y - y^4. Then, integrate the formula for the volume of a solid of revolution with respect to x.

[ V_x = \int_{a}^{b} \pi (f(x))^2 , dx ]

In this case, (f(x) = y - y^4). To find the limits of integration, set the equations equal to each other and solve for y.

[ y - y^4 = 0 ] [ y(1 - y^3) = 0 ] [ y = 0 \text{ or } y = 1 ]

So, the limits of integration are from y = 0 to y = 1. The integral becomes:

[ V_x = \pi \int_{0}^{1} (y - y^4)^2 , dy ]

After integrating, you can find the volume of the solid of revolution about the x-axis.

b) For rotation about the y-axis, sketch the region bounded by the curves x = 0 and x = y - y^4. Then, integrate the formula for the volume of a solid of revolution with respect to y.

[ V_y = \int_{c}^{d} \pi (f(y))^2 , dy ]

In this case, (f(y) = y - y^4). To find the limits of integration, determine the y-values at which the curves intersect.

[ y - y^4 = 0 ] [ y(1 - y^3) = 0 ] [ y = 0 \text{ or } y = 1 ]

So, the limits of integration are from y = 0 to y = 1. The integral becomes:

[ V_y = \pi \int_{0}^{1} (y - y^4)^2 , dx ]

After integrating, you can find the volume of the solid of revolution about the y-axis.

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