Question about Volume of Solids

In summary, the conversation is about finding the volume of solids that are not solids of revolution. The first problem involves finding the volume of the remaining portion of a sphere after a hole of radius r is bored through its center. The second problem involves finding the volume of a solid with an elliptical base and isosceles right triangle cross-sections perpendicular to the x-axis. The approach for the first problem is to use solid of revolution, while for the second problem, it is similar to integrating solid of revolution but with triangles instead of circles.
  • #1
Gauss177
38
0
I'm having trouble with problems where you have to find the volume of a solids that are not solids of revolution. Can someone help me with these problems and also tell me a general way of approaching these problems? Thanks

Homework Statement


A hole of radius r is bored through the center of a sphere of radius R > r. Find the volume of the remaining portion of the sphere.

2. Homework Statement
The base of S (the solid) is an elliptical region with boundary curve [tex]9x^2 + 4y^2 = 36[/tex]. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.
 
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  • #2
For 1, that is a solid of revolution, try drawing it. For 2, it is very similar to integrating solid of revolution, only instead of adding up circles you are adding up triangles. Again, if you can picture the shape it shouldn't be too hard.
 

What is the definition of volume of a solid?

The volume of a solid is the amount of space that the solid occupies. It is measured in cubic units, such as cubic meters or cubic feet.

How is the volume of a regular solid calculated?

The volume of a regular solid, such as a cube or cylinder, is calculated by multiplying the length, width, and height of the solid. The formula for volume is V = lwh.

What is the difference between volume and surface area of a solid?

Volume is the measure of space occupied by a solid, while surface area is the measure of the total area of the surface of a solid. Volume is measured in cubic units, while surface area is measured in square units.

How is the volume of an irregular solid calculated?

The volume of an irregular solid can be calculated by using different methods, depending on the shape of the solid. One method is to submerge the solid in water and measure the amount of water it displaces. This is known as the displacement method. Another method is to break the solid into smaller, regular shapes and calculate the volume of each shape separately.

Why is volume an important concept in science?

Volume is an important concept in science because it helps us understand the amount of space that a substance or object occupies. It is useful for measuring and comparing the sizes of different objects, as well as for determining the amount of a substance in a given space. Volume is also an important factor in many scientific calculations and experiments.

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