volume

**mikegar813** · April 22nd 2009, 10:59 AM

Find the volume of the solid whose base is the region between the x-axis and the inverted parabola f(x)=4-x^2 if every vertical cross sections of the solid perpendicular to the y-axis are semicircles.

**skeeter** · April 22nd 2009, 02:30 PM

Originally Posted by mikegar813

Find the volume of the solid whose base is the region between the x-axis and the inverted parabola f(x)=4-x^2 if every vertical cross sections of the solid perpendicular to the y-axis are semicircles.

$V = \int_c^d A(y) \, dy$

$y = 4-x^2$

$x = \sqrt{4-y}$

radius of each semicircular cross section is $x$

$A = \frac{\pi}{2} x^2 = \frac{\pi}{2}(\sqrt{4-y})^2 =$

$V = \frac{\pi}{2}\int_0^4 4-y \, dy$

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