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Determining Volumes by Slicing. Learning Objectives. Determine the volume of a solid by integrating a cross-section (the slicing method). Find the volume of a solid of revolution using the disk method. Find the volume of a solid of revolution with a cavity using the washer method.
Learn what a cross section of a solid is, and how to identify what shape is formed from the a plane intersecting a solid. This intersection is known as a cross section and creates a shape. In...
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- Ryne Roper
A solid of revolution is a three-dimensional figure obtained by rotating a two-dimensional figure (or curve) around a straight line (called the axis) that lies in the same plane. An example of a rotation about a line is the concept of a revolving door.
Learn the names of common solid figures, the parts of those solids, and how we describe cross-sections of them.
- because math is complicated
- _A polyhedron is a solid figure where every surface is a polygon. Prisms and pyramids are examples of polyhedra._
- The apex is used to find the height of the 3D pyramid or cone. In a skewed cone or pyramid, the apex does not have to "above" the figure, so the he...
- When the plane (what you are slicing with) is paralell to the base
- A apex is the point that is farthest away from the base of an object.
Sketching and Describing Solids of Revolution. A solid of revolution is a three-dimensional fi gure that is formed by rotating a two-dimensional shape around an axis. The line around which the shape is rotated is called the axis of revolution.
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If we take the part of the curve between 𝑥 = 𝑎 and 𝑥 = 𝑏 and rotate it about the 𝑥 -axis by a complete revolution (i.e., by 3 6 0 ∘ or 2 𝜋), the curve will map out the surface of a solid as it rotated; this is called a solid of revolution, as depicted in the diagram.
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A hemisphere is the half sphere formed by a plane intersecting the center of a sphere. The cut-line forming a hemisphere is a great circle. Volume of a Sphere: Note: The volume of a sphere is actually the volume of the solid inside a sphere, often referred to as a spherical solid.
