Begin by sketching the region bounded by the graph of and the. For example, the right cylinder in Figure 3.(a) is generated by translating a circular region along the \(x\)-axis for a certain length \(h\text\) Every cross-section of the right cylinder must therefore be circular, when cutting the right cylinder anywhere along length \(h\) that is perpendicular to the \(x\)-axis. By revolving the rectangles about the x-axis, you obtain n circular disks, each with a. For now, we are only interested in solids, whose volumes are generated through cross-sections that are easy to describe. Cross-section.Ī cross-section of a solid is the region obtained by intersecting the solid with a plane.Įxamples of cross-sections are the circular region above the right cylinder in Figure 3.(a), the star above the star-prism in Figure 3.(b), and the square we see in the pyramid on the left side of Figure 3.11. Let us first formalize what is meant by a cross-section. Subsection 3.3.1 Computing Volumes with Cross-sections ¶ However, we first discuss the general idea of calculating the volume of a solid by slicing up the solid. For example, circular cross-sections are easy to describe as their area just depends on the radius, and so they are one of the central topics in this section. Generally, the volumes that we can compute this way have cross-sections that are easy to describe.
Circular disk graph how to#
We have seen how to compute certain areas by using integration we will now look into how some volumes may also be computed by evaluating an integral.
Section 3.3 Volume of Revolution: Disk Method ¶ Power Series and Polynomial Approximation.First Order Linear Differential Equations.Triple Integrals: Volume and Average Value.Double Integrals: Volume and Average Value.Partial Fraction Method for Rational Functions.
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