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#