How do you use an integral to find the volume of a solid torus?
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To find the volume of a solid torus using an integral, you can use the formula:
[ V = \pi \int_{r_1}^{r_2} (R^2 - r^2) , dr ]
Where:
- ( V ) is the volume of the torus.
- ( \pi ) is the constant pi.
- ( r_1 ) is the radius of the smaller circle (cross-section).
- ( r_2 ) is the radius of the larger circle (cross-section).
- ( R ) is the distance from the center of the torus to the center of either cross-section.
You integrate from ( r_1 ) to ( r_2 ) because you're summing the volumes of infinitesimally thin disks that make up the torus. Each disk has a radius ( r ) that varies from ( r_1 ) to ( r_2 ), and its volume is given by ( \pi (R^2 - r^2) , dr ).
By evaluating this integral, you can find the volume of the solid torus.
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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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- How do you Evaluate the integral by changing to cylindrical coordinates?

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