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

May 12, 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 around the y-axis, we use the method of cylindrical shells

step 2

The formula for the volume of a cylindrical shell is

V = 2\pi \int_a^b x f(x) \, dx

, where

f(x)

is the function being rotated

step 3

In this case,

f(x) = x^2

and the limits of integration are from

a = 0

b = 2

, since the region is bounded by

y = 4

and

y = x^2

intersects

y = 4

x = 2

step 4

Substituting

f(x) = x^2

into the formula, we get

V = 2\pi \int_0^2 x (x^2) \, dx

step 5

This integral simplifies to

V = 2\pi \int_0^2 x^3 \, dx

step 6

Using the power rule for integration, we find the antiderivative of

x^3

to be

\frac{x^4}{4}

step 7

Evaluating the definite integral from 0 to 2, we get

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

step 8

Plugging in the limits of integration, we find

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

step 9

Simplifying, we get

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

step 10

Therefore, the volume of the solid is

V = 8\pi

Answer

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

y = x^2

x = 0

, and

y = 4

about the y-axis 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 the solid is generated by rotating a region around an axis, in this case, the y-axis. The volume is found by integrating the product of the circumference of the shell, the height of the shell, and the thickness of the shell.