Solids cross section
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This document discusses several methods for calculating the volume of solids with known cross-sectional areas: calculating the volume of prisms and cylinders by multiplying the area of their circular or rectangular base by the height; using the disk method for solids of revolution by multiplying the circular cross-sectional areas by the width; and applying this method to solids with any shaped cross-section, such as squares, triangles, or ellipses, by multiplying the varying cross-sectional area by the width. Formulas for calculating volume by integrating the cross-sectional area with respect to x or y are also provided, along with an example problem.
Solids cross section
- 1. Volumes of solids with known cross sections • You have already seen this being done in geometry. • For example, finding the volume of a prism , or a cylinder: • Find the area of the Base, and mutiply by the width or height.
- 2. The disk and washer method • We have also seen this in our study of Calculus via Solids of revolution • The disk and washer method use Circular cross sections • We multiply the cross sections times the width to get the actual volume.
- 3. It can really be ANY cross section • It doesn’t necessarily have to be circular. • It can be a square, triangle, ellipse, etc.. • If we can find the Area of the cross section, then we can multiple it times the width (just like before) to get the entire volume. • ANIMATION
- 4. Formulas • When the cross section is perpendicular to the x-axis, integrate with respect to x: b V A ( x ) dx a • When the cross section is perpendicular to the y-axis, integrate with respect to y: b V A ( y ) dy a
- 5. Example: • A solid is formed with a base bounded by the graphs of: f ( x) 1 x 2 x 0 g ( x) 1 x 2 Find the volume of the solid using equilateral triangle cross – sections taken perpendicular to the x-axis.
- 6. Practice