#### To determine

**To find:**

The volume of solid S whose base is the region enclosed by the parabola y=1-x2

and the x-axis. Cross sections perpendicular to the y-axis are squares.

#### Answer

V=2

#### Explanation

**1) Concept:**

Definition of volume:

Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is continuous function, then the volume of S is

V=limn→∞∑i=1nAxi*∆x=∫abAxdx

**2) Calculations:**

The (red line segment) represents the side of a square (cross section). Since s is perpendicular to y-axis, write the equation of parabola for in terms of x.

Equation of parabola is

y=1-x2

Solve this equation for x,

Therefore,

x=±1-y

Thus length of s at y is given by 1-y - (-1-y)=21-y.

That is s(y)=21-y.So the cross section at y is a square of side s(y). Thus the area of cross section at y is

A(y)=21-y2=41-y

Therefore, to find volume, integrate cross-section from 0 to 1,

V=∫01A dy

V=∫0141-ydy

After integrating,

V=4(y-y22)10

Applying the Fundamental Theorem of Calculus,

V=41-122-40-0=42=2

**Conclusion:**

The volume of solid S whose base is the region enclosed by the parabola y=1-x2

And the -axis, and cross sections perpendicular to the y-axis are squares is 2