A solid lies between planes perpendicular to the y-axis at [tex]y=0[/tex] and [tex]y=4[/tex]. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola [tex]x=\sqrt{10} y^2[/tex]. Find the volume of the solid.

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16-09-2022
Mathematics

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A solid lies between planes perpendicular to the y-axis at [tex]y=0[/tex] and [tex]y=4[/tex]. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola [tex]x=\sqrt{10} y^2[/tex]. Find the volume of the solid.

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LammettHash LammettHash · Answer 1 · 2022-09-16T20:40:36-04:00

For any given [tex]0\le y\le4[/tex], a cross section in that plane is a circle whose diameter is [tex]x=\sqrt{10}\,y^2[/tex] with thickness [tex]\Delta y[/tex]. Such a cross section contributes a volume of [tex]\pi \left(\frac x2\right)^2 \, \Delta y = \frac{5\pi}2 y^4 \, \Delta y[/tex].

As [tex]\Delta y\to0[/tex] and the number of cross sectional cylindrical disks increases without bound, the infinite sum of their volumes converge to the volume of the solid, given by the definite integral

[tex]\displaystyle \frac{5\pi}2 \int_0^4 y^4 \, dy = \frac{5\pi}2 \cdot \frac{4^5}5 = \boxed{512\pi}[/tex]

A solid lies between planes perpendicular to the​ y-axis at [tex]y=0[/tex] and [tex]y=4[/tex]. The​ cross-sections perpendicular to the​ y-axis are circular disks with diameters running from the​ y-axis to the parabola [tex]x=\sqrt{10} y^2[/tex]. Find the volume of the solid.

Answer :

Other Questions

A solid lies between planes perpendicular to the y-axis at [tex]y=0[/tex] and [tex]y=4[/tex]. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola [tex]x=\sqrt{10} y^2[/tex]. Find the volume of the solid.