#### To determine

**To find:**

The volume of solid S whose base is the region enclosed by y=2-x2 and x-axis and cross sections are perpendicular to the y-axis are quarter circles.

#### Answer

2π

#### 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 cross sections are perpendicular to the y-axis.

Therefore, solve the equation of parabola for x.

y=2-x2

x= ±2-y

So the perpendicular distance of a point from y-axis is 2-y. So, to get length of base of cross section double the distance from above.

Thus cross section of solid S at y is a quarter ciecle with radius 22-y.

Area of circle is πr2.

Therefore, area of a quarter circle is 14πr2.

Ay=14π22-y2

After simplification,

Ay=π(2-y)

Therefore, to find volume, integrate Ay from 0 to 2,

V=∫02π2-ydy

After integrating,

V=π2y-y222 0

By using the limits,

V=π2·2-222-0-0=2π

**Conclusion:**

The volume of solid S whose base is the region enclosed by y=2-x2 and x-axis and cross sections perpendicular to the y-axis are quarter circles is 2π