# 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.

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- How do you find the volume of a solid that is enclosed by #y=2x+2# and #y=x^2+2# revolved about the x-axis?
- Calculate the volume of a solid whose base is the ellipse # 4x^2 + y^2 = 4 # and has vertical cross sections that are square?
- How do you Evaluate the integral by changing to cylindrical coordinates?

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