   Chapter 5.R, Problem 34E

Chapter
Section
Textbook Problem

# Let ℜ 1 be the region bounded by y = x 2 , y = 0 , and x = b , where b>0. Let ℜ 2 be the region bounded by y = x 2 , x = 0 , , and y = b 2 .(a) Is there a value of b such that ℜ 1 and ℜ 2 have the same area?(b) Is there a value of b such that ℜ 1 sweeps out the same volume when rotated about the x -axis and the y -axis?(c) Is there a value of b such that ℜ 1 and ℜ 2 sweep out the same volume when rotated about the x -axis?(d) Is there a value of b such that ℜ 1 and ℜ 2 sweep out the same volume when rotated about the y -axis?

To determine

(a)

To find:

A value of b such that R1 and R2 have the same area.

Explanation

1) Concept:

Area of the region is the same as the integration below the curve for the given interval.

2) Given:

R1 is bounded by y=x2, y=0 and x=b, where b>0.

R2 is bounded by y=x2, x=0 and y=b2, where b>0.

3) Calculation:

R1 is the region below y=x2 and above x-axis and in between x=0 and x=b.

R2 is the region to the left of y=x2 and to the right of y-axis and in between y=0 and y=b2.

So the area of R1 is

A1=0bx2dx

=x33b0

To determine

(b)

To find:

A value of b  such that R1 sweeps out the same volume when rotated about x-axis and

y-axis.

To determine

(c)

To find:

A value of b such that R1 and R2 sweep out the same volume when rotated about x-axis.

To determine

(d)

To find:

A value of b such that R1 and R2 sweeps out same volume when rotated about y-axis.

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