Help with Volume circular disk.

In summary, the solid S has a circular base with radius 3r and parallel cross-sections perpendicular to the base that are squares. To find the volume V of this solid, you can integrate the area A(x) of each cross-section from x = 0 to x = 3r and multiply by 2 due to the symmetry of the solid. The length of a chord of the circle with its midpoint at a distance x from the circle's center will give you the area of each cross-section.
  • #1
quickclick330
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0
Consider the solid S described below.
The base of S is a circular disk with radius 3r. Parallel cross-sections perpendicular to the base are squares.
Find the volume V of this solid.


I tried this...


A(x) = pi*r^2 = pi*(3r)^2 = pi*9r^2

V(x) = integral from 0 to 3r(pi*9r^2 dx) = pi*27r^3


Am I approaching this problem the wrong way? Thanks for the help!
 
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  • #2
quickclick330 said:
Consider the solid S described below.
The base of S is a circular disk with radius 3r. Parallel cross-sections perpendicular to the base are squares.
Find the volume V of this solid.

The square cross-sections are built upon the circular base, so each cross-section has a side length given by the length of a chord of the circle; these "slices" are standing upright on the circle. (Making a picture of this will be helpful to you.) The solid is symmetric about a diameter of the circle, so you can integrate half the solid and multiply your result by two.

So what you need is to find the length of a chord of the circle which has its midpoint at a distance x from the circle's center. That length gives you the area A(x) of each cross-section. You would then integrate A(x) from x = 0 to x = 3r and multiply the resulting volume by 2.
 

1. What is the formula for calculating the volume of a circular disk?

The formula for calculating the volume of a circular disk is V = πr^2h, where r is the radius of the disk and h is the height or thickness of the disk.

2. How do I find the radius of a circular disk?

To find the radius of a circular disk, you can measure the distance from the center of the disk to the edge using a ruler or measuring tape. Alternatively, if you know the diameter of the disk, you can divide it by 2 to get the radius.

3. Can the height or thickness of a circular disk affect the volume?

Yes, the height or thickness of a circular disk can affect the volume. The volume of a disk is directly proportional to its height, meaning that as the height increases, the volume also increases.

4. How do I convert the volume of a circular disk from cubic units to other units of measurement?

To convert the volume of a circular disk from cubic units to other units of measurement, you can use conversion factors. For example, to convert from cubic centimeters to milliliters, you would multiply the volume in cubic centimeters by 1000.

5. Can I use the formula for calculating the volume of a circular disk for other shapes?

No, the formula for calculating the volume of a circular disk is specific to circular disks and cannot be used for other shapes. Each shape has its own formula for calculating volume.

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