Solids cross section

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Information about Solids cross section

Published on February 21, 2014

Author: roneick


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.

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.

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

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

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.


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