# The region under the curves #x, y>=0, y=x^2sqrt(1-x^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, you first need to find the points of intersection of the curves (y = x) and (y = x^2\sqrt{1 - x^4}). Then, you need to determine the interval on which the curves intersect. After that, sketch the curves within this interval.

To find the volume of the solid of revolution when rotated about the x-axis, use the formula for the volume of a solid of revolution: (V = \int_{a}^{b} \pi [f(x)]^2 dx), where (a) and (b) are the x-values of the intersection points, and (f(x)) is the function representing the upper curve.

To find the volume of the solid of revolution when rotated about the y-axis, you use the formula: (V = \int_{c}^{d} \pi [f(y)]^2 dy), where (c) and (d) are the y-values of the intersection points, and (f(y)) is the function representing the rightmost curve.

After setting up the integrals, you integrate them to find the volumes.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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