$Explain in detail how you use triple integrals to find the volume of the solid.$

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$Explain in detail how you use triple integrals to find the volume of the solid.$

The solid generated by rotating the curve $r(t) = (t^2,t^3- t) $ with $t \in [-1,1]$ around the y-axis.

Multivariable Calculus Integrals Calculus

Rafalpha

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Kav10

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Using cylindrical coordinates $(\rho, \phi, y)$ where $\rho$ is the distance from the y-axis, I get
volume $V = 2 \pi \int_0^1 ( 2 \eta(\rho) ) \rho \,\mathrm d\rho = 4 \pi ( 2/5 - 2/7) = 16\pi/35 $,
where $\eta(x)= (1-x)\sqrt x$ is the distance of the point $r(t) = (t^2,t^3-t)$ from the x-axis when the distance from the y-axis is x:

See the attached file for more details and a graphics.

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M F H

Rafalpha

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When you write "t=η (x(t))sign(t)", you mean "t=η (x(t))sin(t)" ?
M F H

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No, sign(t) is the sign of t, -1 when t < 0 and +1 when t > 0. This is to get back the "full t" including its sign, when you take the square root of the square (which would otherwise always be positive).
M F H

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Thanks for the coffee! 🙂

$Explain in detail how you use triple integrals to find the volume of the solid.$

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