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A glass is formed by rotating the shaded region shown above about the

y

axis. ...

Feb 1, 2024

A glass is formed by rotating the shaded region shown above about the

y

axis. The curve that forms the inside of the glass is the graph of

y=x^{4} / 2

. Length units in the figure are

\mathrm{cm}

. (a) What is the volume of the glass? (That is, what is the volume of the solid formed when the shaded region is rotated about the

y

axis?)

Solution by Steps

step 1

To find the volume of the glass, we use the method of disks to integrate along the

y

-axis. The volume

V

of the solid of revolution is given by the integral

V = \int_{a}^{b} \pi [f(x)]^2 dx

, where

f(x)

is the function being rotated

step 2

The function given is

y = \frac{x^4}{2}

. When rotated about the

y

-axis, the radius of the disk at a given

x

x

, and the area of the disk is

\pi x^2

. The volume element is then the area of the disk times the thickness

dx

, which is

\pi x^2 dx

step 3

The volume of the glass is the integral of the volume element from

x=0

x=h

. Thus,

V = \int_{0}^{h} \pi x^2 \left(\frac{x^4}{2}\right) dx

step 4

Simplifying the integrand, we have

V = \int_{0}^{h} \pi \frac{x^6}{2} dx

step 5

Using the power rule for integration, we integrate

x^6

to get

\frac{x^7}{7}

step 6

The definite integral from

0

h

gives us

V = \left[\frac{\pi}{2} \frac{x^7}{7}\right]_{0}^{h}

step 7

Evaluating the integral from

0

h

, we find

V = \frac{\pi}{2} \frac{h^7}{7} - \frac{\pi}{2} \frac{0^7}{7}

step 8

Simplifying, we get

V = \frac{\pi h^7}{14}

Answer

The volume of the glass is

\frac{\pi h^7}{14}

cubic centimeters.

Key Concept

Volume of a solid of revolution using the disk method

Explanation

The volume of a solid formed by rotating a function

y=f(x)

about the

y

-axis from

x=a

x=b

is found by integrating

\pi [f(x)]^2

with respect to

x

from

a

b