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Compute the volume of the solid obtained by rotating the region bounded by $y = ...

Apr 18, 2024

Compute the volume of the solid obtained by rotating the region bounded by

y = x^2

x = 0

, and

y = 4

about the y-axis.

Generated Graph

Solution by Steps

step 1

To find the volume of the solid obtained by rotating the region bounded by

y = x^2

x = 0

, and

y = 4

about the y-axis, we use the method of cylindrical shells

step 2

The volume of a thin cylindrical shell with radius

r

, height

h

, and thickness

\Delta x

is given by

2\pi rh \Delta x

step 3

For the given region, the radius of the shell is

r = x

and the height is

h = x^2

step 4

The volume of the solid is the integral of the volume of the shells from

x = 0

x = 2

, which is the boundary of the region for

y = x^2

when

y = 4

step 5

The integral for the volume is

V = \int_{0}^{2} 2\pi x(x^2) dx

step 6

Simplify the integral to

V = 2\pi \int_{0}^{2} x^3 dx

step 7

Using the power rule for integration, we find

V = 2\pi \left[\frac{x^4}{4}\right]_0^2

step 8

Evaluate the integral from

0

2

to get

V = 2\pi \left[\frac{2^4}{4} - \frac{0^4}{4}\right]

step 9

Simplify to find

V = 2\pi \left[\frac{16}{4}\right] = 2\pi [4] = 8\pi

Answer

The volume of the solid is

8\pi

cubic units.

Key Concept

Method of Cylindrical Shells

Explanation

The method of cylindrical shells is used to compute the volume of a solid of revolution when integrating along the axis of rotation. The volume is found by integrating the surface area of cylindrical shells.