Volume by Cylindrical Shells

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2015-06-09 21:55

This example demonstrates how to calculate the volume of a solid of revolution around a line other than an axis using the method of cylindrical shells. Key math concepts include integral calculus, volume calculation, and solid geometry. GeoGebra commands and objects like LaTeX, Point, Segment, Mirror, and variables such as a, b, f, g are used to visualize the rotation and volume computation. Suitable for high school students and above to learn about volume calculation techniques for solids of revolution.

GeoGebra Commands & Parameters

LaTeX(("\text{step }" * steps)) → text1
LaTeX("\text{The general form of the integrand is } 2\pi(radius)(height)\text{.}") → text3
Point(xAxis) → A
Segment((x(A), g(x(A))), (x(A), f(x(A)))) → h
Point(h) → B
Segment(B, (2, y(B))) → r
LaTeX("\text{In this case the radius is given by }\textcolor{#FF33CC}{r}=x-\textcolor{#CC0000}{2}\text{.}") → text4
LaTeX("\text{The integration limits come from the x-coordinates of the}\cr\text{intersection points of functions }\textcolor{#000099}{f}\text{ and }\textcolor{#339900}{g}.") → text6
LaTeX((((((("\text{Use cylindrical shells to calculate the volume swept out when the}\cr\text{region enclosed by the functions }\textcolor{#0033CC}{f=" * f) * "}\text{ and }\textcolor{#339900}{g=e^{") * a) * "x}}\text{ is}\cr\text{revolved around the line }\textcolor{#CC0000}{") * c) * "}.")) → text2
LaTeX("\text{The height }\textcolor{#FF6600}{h}\text{ is given by }\textcolor{#0033CC}{f(x)}-\textcolor{#339900}{g(x)}.") → text5
LaTeX((((((((("V=2\pi\int_{" * Lintersect) * "}^{") * Rintersect) * "}(x-2)\left[(") * f) * ")-(") * g) * ")\right]dx")) → text7
Mirror(fcut, c) → fshadow
Mirror(gcut, c) → gshadow
Mirror(h, c) → h'
Mirror(r, c) → r'