   Chapter 5.3, Problem 8E

Chapter
Section
Textbook Problem

# Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by y = x and y = x 2 . Find V both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.

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.

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,  0ab

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

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=πy

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