Volume of a Torus Using Pappus's Theorem

The video tutorial explains how to use Pappus's Theorem to calculate the exact volume of a torus. It begins by setting up the problem with a circle of radius 3 inches, positioned 8 inches from the y-axis. The theorem is explained, highlighting the need to calculate the area of the circle and the distance traveled by its centroid when rotated around the y-axis. The area is calculated as 9π square inches, and the distance as 22π inches. Finally, the volume of the torus is determined to be 198π² cubic inches, with an approximate volume of 1954.18 cubic inches.

What is the radius of the circle used to form the torus?

• 2 inches

• 5 inches

• 3 inches

• 4 inches

Where is the left end of the circle positioned relative to the y-axis?

• 7 inches

• 8 inches

• 5 inches

• 6 inches

According to Pappus's Theorem, what must be multiplied to find the volume of a solid of revolution?

• Radius and height

• Circumference and radius

• Diameter and height

• Area of the region and distance traveled by the centroid

What is the formula for the area of a circle?

• 2πr

• πr²

• πd

• 2πr²

What is the area of the circle used in this problem?

• 6π square inches

• 8π square inches

• 7π square inches

• 9π square inches

What is the x-coordinate of the centroid of the circle?

• 8

• 11

• 10

• 9

What is the distance traveled by the centroid when the circle is rotated around the y-axis?

• 22π inches

• 21π inches

• 20π inches

• 23π inches

What is the formula for the circumference of a circle?

• πr²

• 2πr

• 2πd

• πd

What is the exact volume of the torus?

• 199π cubic inches

• 199π² cubic inches

• 198π² cubic inches

• 198π cubic inches

What is the approximate volume of the torus?

• 1956.18 cubic inches

• 1952.18 cubic inches

• 1954.18 cubic inches

• 1950.18 cubic inches