# The region under the curves #y=x^3, y=x^2# 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) When rotated about the x-axis:

- Sketch the curves y = x^3 and y = x^2 on the coordinate plane.
- Identify the region enclosed by these curves.
- To find the volume of the solid generated by rotating this region about the x-axis, use the disk/washer method.
- Set up the integral using the formula for the volume of a solid of revolution: V = ∫[a, b] π(R^2 - r^2) dx, where R is the outer radius and r is the inner radius.
- The outer radius (R) is the distance from the axis of rotation (x-axis) to the curve y = x^3, and the inner radius (r) is the distance from the axis of rotation to the curve y = x^2.
- Evaluate the integral to find the volume of the solid.

b) When rotated about the y-axis:

- Repeat steps 1 and 2 above to sketch the curves and identify the enclosed region.
- To find the volume of the solid generated by rotating this region about the y-axis, also use the disk/washer method.
- Set up the integral using the formula for the volume of a solid of revolution: V = ∫[c, d] π(R^2 - r^2) dy, where R is the outer radius and r is the inner radius.
- In this case, the outer radius (R) is the distance from the axis of rotation (y-axis) to the curve x = (y)^(1/2), and the inner radius (r) is the distance from the axis of rotation to the curve x = (y)^(1/3).
- Evaluate the integral to find the volume of the solid.

That's how you sketch the region and find the volumes of the two solids of revolution.

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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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