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?

Answer 1

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#
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Answer 2

To 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, you can use the method of slicing and integration.

The cross sections perpendicular to the y-axis will be squares with side length equal to the corresponding y-coordinate on the curve.

The volume (V) can be calculated by integrating the area of each square cross section over the range of y-values from 0 to 1 (the range of the curve).

The area of each square cross section is y^2, and integrating this over the range of y-values from 0 to 1 gives the volume:

V = ∫[0 to 1] y^2 dy

After evaluating this definite integral, you'll find the volume of the solid bounded by the curve y=1-x^2, the x-axis, and the cross sections perpendicular to the y-axis.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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