How do you use an integral to find the volume of a solid torus?

Answer 1
If the radius of its circular cross section is #r#, and the radius of the circle traced by the center of the cross sections is #R#, then the volume of the torus is #V=2pi^2r^2R#.
Let's say the torus is obtained by rotating the circular region #x^2+(y-R)^2=r^2# about the #x#-axis. Notice that this circular region is the region between the curves: #y=sqrt{r^2-x^2}+R# and #y=-sqrt{r^2-x^2}+R#.
By Washer Method, the volume of the solid of revolution can be expressed as: #V=pi int_{-r}^r[(sqrt{r^2-x^2}+R)^2-(-sqrt{r^2-x^2}+R)^2]dx#, which simplifies to: #V=4piR\int_{-r}^r sqrt{r^2-x^2}dx# Since the integral above is equivalent to the area of a semicircle with radius r, we have #V=4piRcdot1/2pi r^2=2pi^2r^2R#
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Answer 2

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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Answer from HIX Tutor

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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