Question about Volume of Solids

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SUMMARY

The discussion focuses on calculating the volume of solids that are not solids of revolution, specifically addressing two problems: finding the volume of a sphere with a cylindrical hole and determining the volume of a solid with an elliptical base and triangular cross-sections. The first problem involves a sphere of radius R with a hole of radius r, while the second problem requires integrating the area of isosceles right triangles perpendicular to the x-axis within an elliptical boundary defined by the equation 9x² + 4y² = 36. A visual understanding of the shapes involved is crucial for solving these problems effectively.

PREREQUISITES
  • Understanding of volume calculation for three-dimensional shapes
  • Familiarity with integration techniques in calculus
  • Knowledge of geometric properties of ellipses and triangles
  • Ability to visualize cross-sections of solids
NEXT STEPS
  • Study the method for calculating the volume of solids with holes, specifically using the formula for the volume of a sphere.
  • Learn how to set up and evaluate integrals for solids with non-standard cross-sections.
  • Explore the properties of isosceles right triangles and their application in volume calculations.
  • Practice problems involving elliptical regions and their corresponding volumes.
USEFUL FOR

Students studying calculus, particularly those focusing on volume calculations of irregular solids, as well as educators seeking to enhance their teaching methods for geometric volume problems.

Gauss177
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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 9x^2 + 4y^2 = 36. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.
 
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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.
 

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