#### To determine

**To find:**

The volume of the solid obtained by rotating the region bounded by the given curves about y-axis, using the cylindrical shells and slicing.

#### Answer

i. By the slice method

V=3π10unit3

ii. By the shell method

V=3π10unit3

#### Explanation

**1) Concept:**

i. If x is the radius of the typical shell, then the circumference =2πx and the height is y

ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about y-axis from a to b is

V= ∫ab2πx f(x)dx

Where, 0≤a≤b

iii. If the cross section is the disc and the radius of the disc is in terms of x or y, then

A=π radius2

iv. The volume of the solid revolution about the y-axis,

V= ∫abA(y)dy

**2) Given:**

The region bounded by y=x and y=x2 rotated about the y- axis.

**3) Calculation:**

As region is bounded by y=x and y=x2 rotated about the y- axis.

i. By the slice method,

From the figure, as the region rotates about y –axis strip is perpendicular to y-axis.

Find the intersection of the two curves by solving the simultaneous equation,

x=x2

x-x2=0

x=0 and x=1

Therefore, the curves y=x and y=x2 intersects at point (0, 0) and (1, 1)

A cross section has the shape of a washer with the inner radius x=y2 and the outer radius x=y

So, find the area of cross-sectional by subtracting the area of the inner circle and the outer circle:

Ay= πy2-πy4

Ay=πy-y4

Therefore, the volume of the solid of revolution about y-axis,

V= ∫01Aydy=∫01πy-y4dy

By using the fundamental theorem of calculus and the power rule of integration,

V=πy22-15y501

=π 122-1515-0

=π12-15

=π 310

V=3π10

ii. By shell method,

Find the typical approximating shell with the radius x as shown below.

Therefore, the circumference is 2πx and the height is x-x2

To find a and b, consider x=x2

x=0 or x=1

Therefore, a=0 and b=1

So, the total volume is

V= ∫ab2πx[x -x2] dx

V=2 π∫01[x32-x3] dx

V=2π2x525-x4401

=2π 2(1)525-144-0

=2π25-14

=2π320

V=3π10

**Conclusion:**

The volume of the solid obtained by rotating the region bounded by the given curves is

i. By the slice method

V=3π10unit3

ii. By the shell method

V=3π10unit3

That is, the volume is same for both the methods.