The region under the curves #y=x^-2, 1<=x<=2# is rotated about the x axis. How do you find the volumes of the two solids of revolution?
Ezekiel AlexanderSee the answer below. In fact, rotating about the x axis will produce only one solid.
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Ezra AdamsTo find the volumes of the two solids of revolution formed by rotating the region under the curves y=x^(-2), 1≤x≤2 about the x-axis, you can use the method of cylindrical shells.
- First, find the volume of the solid formed by rotating the region under the curve y=x^(-2) from x=1 to x=2.
- Then, find the volume of the hole (or void) that is formed in the center when the solid is removed.
To find the volume of the solid:
- Integrate from 1 to 2 the expression 2πx * (x^(-2)) * dx.
To find the volume of the void:
- Integrate from 1 to 2 the expression 2πx * (x^(-2)) * dx.
Subtract the volume of the void from the volume of the solid to get the final volume of the solid 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.
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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