AskSia

Plus

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

Apr 7, 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 r

is given by

2\pi rh \Delta r

step 3

In this case, the radius

r

x

, the height

h

y = x^2

, and

\Delta r

dx

step 4

The volume

V

is the integral of the volume of the shells from

x = 0

x = 2

(since

y = 4

when

x = 2

)

step 5

Set up the integral for the volume:

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

step 6

Simplify the integral:

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

step 7

Calculate the integral using the power rule:

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

step 8

Evaluate the integral from

x = 0

x = 2

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

step 9

Simplify to find the volume:

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

Answer

V = 8\pi

cubic units

Key Concept

Cylindrical Shells Method

Explanation

The volume of the solid of revolution is calculated using the cylindrical shells method, which involves integrating the volume of infinitesimally thin cylindrical shells between the given bounds.