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

Dec 29, 2023

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.

Solution by Steps

step 1

Identify the method of calculating the volume of the solid of revolution. For this problem, we use the disk method, which involves integrating the area of circular disks along the axis of rotation

step 2

Set up the integral for the volume. The volume

V

of the solid formed by rotating the region about the y-axis from

y = 0

y = 4

is given by the integral

V = \pi \int_{0}^{4} [r(y)]^2 dy

where

r(y)

is the radius of the disk at height

y

step 3

Express

x

as a function of

y

since we are rotating around the y-axis and the region is bounded by

y = x^2

. Thus,

r(y) = x = \sqrt{y}

step 4

Substitute

r(y)

into the integral and evaluate it:

V = \pi \int_{0}^{4} (\sqrt{y})^2 dy = \pi \int_{0}^{4} y dy

step 5

Compute the integral:

V = \pi \left[ \frac{y^2}{2} \right]_{0}^{4} = \pi \left( \frac{4^2}{2} - \frac{0^2}{2} \right) = \pi \left( \frac{16}{2} \right) = 8\pi

Answer

The volume of the solid is

8\pi

cubic units.

Key Concept

Disk method for volumes of solids of revolution

Explanation

The disk method involves integrating the area of circular disks perpendicular to the axis of rotation to find the volume of the solid.