How do you find the volume of the solid with base region bounded by the curve $y=1-x^2$ and the $x$ -axis if cross sections perpendicular to the $y$ -axis are squares?

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Answer 1 · 2014-09-22T03:58:53

The volume is 2.

Let us look at some details.

By rewriting,

$y=1-x^2 Leftrightarrow x=pmsqrt{1-y}$

Since the length of the side of each cross sectional square is $2sqrt{1-y}$ , the cross sectional area $A(y)$ can be given by

$A(y)=(2sqrt{1-y})^2=4(1-y)$

Since the base spans from $y=0$ to $y=1$ , the volume $V$ can be found by

$V=4int_0^1(1-y)dy=4[y-y^2/2]_0^1=4(1-1/2)=2$

How do you find the volume of the solid with base region bounded by the curve y=1-x^2y=1-x^2 and the xx-axis if cross sections perpendicular to the yy-axis are squares?