Respuesta :

LammettHash
Using shells, the volume is

[tex]\displaystyle2\pi\int_0^6(6-x)x\,\mathrm dx[/tex]

where [tex]6-x[/tex] is the distance from any point in the region along the x-axis to the axis of revolution [tex]x=6[/tex], which makes up the radius of each shell; and [tex]x[/tex] is the height of each shell, which is determined by the line [tex]y=x[/tex].

So the volume is

[tex]\displaystyle2\pi\int_0^6(6x-x^2)\,\mathrm dx=2\pi\left(3x^2-\frac{x^3}3\right)\bigg|_{x=0}^{x=6}=72\pi[/tex]

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