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Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the $y$ -axis.$$y=x^{3}, x=1, y=0$$
$\mathbf{V}=\frac{2 \pi}{5}$
Calculus 1 / AB
Calculus 2 / BC
Chapter 6
APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING
Section 3
Volumes by Cylindrical Shells
Integrals
Integration
Applications of Integration
Area Between Curves
Volume
Arc Length and Surface Area
Campbell University
Oregon State University
Harvey Mudd College
University of Nottingham
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this problem. We want to use cylindrical shelves to find the volume of a solid generated when the curve like fools X cubed is revolved around the why access And we know we're looking at How so? We also need the curve XY pulled one. So what is this picture look like? So we have and started and white zero. So we'll use blue for this x cubed That looks like this. Oh my gosh, I can't drive it. Here we go and we screen for X equals one. And then what Use red for by equals zero That's here. So then shape it area between it's gonna be right here That's getting rotated about the Why access. So from here we need So in this problem, we're kind of demonstrating how we take a region, then apply swim turtle shells to find the volume we were to rotate their agent about a specific access stomach and set ups. We have V is going to equal the integral. So X goes from 0 to 2. And here's my age. My height. So you go from zero sorry. 01 cash from 012 pi X and what's f of X So f of X is going to be X cube because that's a curb on top, minus zero. That's the curve on the bottom DX. So then we're gonna take the two pi, bring it in front of the inner role, distribute this X plus X to the fourth DX, so it's gonna be two pi times 1/5 exit fifth evaluated from 0 to 1. So that's gonna be two bits high that complete.
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