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The base of S is the region enclosed by the parabola y = 4 − 2x2 and the x−axis. Cross-sections perpendicular to the y−axis are squares. FIND THE VOLUME

Respuesta :

LammettHash
The length of each cross-section is determined by the horizontal distance (parallel to the x-axis) from one end of the parabola to the other.

[tex]y=4-2x^2\implies x=\pm\sqrt{\dfrac{4-y}2}[/tex]

which means this horizontal distance is

[tex]\sqrt{\dfrac{4-y}2}-\left(-\sqrt{\dfrac{4-y}2}\right)=2\sqrt{\dfrac{4-y}2}[/tex]

The area of each cross-section is simply the square of the section's side length, so the area would be [tex]\left(2\sqrt{\dfrac{4-y}2}\right)^2=4\dfrac{4-y}2=8-2y[/tex].

So the volume of the solid would be

[tex]\displaystyle\int_0^4(8-2y)\,\mathrm dy=16[/tex]

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