#### To determine

**(a)**

**To find:**

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

#### Answer

No any value for b.

#### 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

=13b3

And area of R2 is

A2=∫0b2y dy

=y3232b20

=23b3

So, for A1=A2,

That is, for 13b3=23b3

Solve for b3

13b3=0

Therefore, b equal to zero.

But b>0

Therefore,

There is no value of b≠0 such that R1 and R2 have the same area.

**Conclusion:**

There is no value of b≠0 such that R1 and R2 have the same area.

#### 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.

#### Answer

b=52

#### Explanation

**1) Concept:**

Equate the volumes obtained by rotating R1 about x-axis and rotatingabout y-axis to calculate b.

**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**:

By using disk method, find the volume of solid formed by rotation of R1 about x-axis.

V1=∫0bπx22 dx

=πb55

By using cylindrical shells, find the volume of rotation of R1 about y-axis.

V2=2π∫0bxx2 dx

=2πb44

=π2b4

To find the value of b, equate V1 and V2.

Therefore,

πb55=π2b4

Solving for b,

b=52

Therefore, two volumes are the same for b=52.

**Conclusion:**

R1 sweeps out the same volume when rotated about x-axis and y-axis when b=52.

#### To determine

**(c)**

**To find:**

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

#### Answer

No any value of b≠0.

#### Explanation

**1) Concept:**

Equate the volumes obtained by rotating R1 and R2 about x-axis to calculate b.

**2) Calculation:**

From part (b),

The volume of rotation of R1 about x-axis is

V1=∫0bπx22dx=πb55

Now, using cylindrical shells,

The volume of rotation of R2 about x-axis,

V2=2π∫0b2yy dy=2π25y52b20=4πb55

To find the value of b,

Equate V1 and V2.

Therefore, πb55=4πb55

This two volumes are equal only if b=0.

Therefore, there is no b≠0 such that V1=V2.

**Conclusion:**

There is no value of b≠0 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.

#### Answer

For all values of b, R1 and R2 sweeps out same volume when rotated about y-axis.

#### Explanation

**1) Concept:**

Equate the volumes obtained by rotating R1 and R2 about y-axis to calculate b.

From part (b),

The volume of rotation of R2 about y-axis is

V2=2π∫0bxx2dx=2πb44=π2b4

Now, using disks,

The volume of rotation of R1 about y-axis is

V1= π∫0b2y2dy=πy22b20=π2b4

Equating V1 and V2,

π2b4=π2b4

which is true for all b.

**Conclusion:**

For all values of b,R1 and R2 sweep out same volume when rotated about y-axis.