   Chapter 5.2, Problem 60E

Chapter
Section
Textbook Problem

# Find the volume of the described solid S.The base of S is the region enclosed by y = 2 − x 2 and the x-axis. Cross-sections perpendicular to the y-axis are quarter-circles. 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.

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=limni=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

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