AskSia

Plus

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

Dec 24, 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

To 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, we use the disk method. The volume

V

of the solid is given by the integral

V = \pi \int_{a}^{b} [R(y)]^2 dy

, where

R(y)

is the radius of the disk at height

y

step 2

The region is bounded by

y = x^2

and

y = 4

. Solving for

x

in terms of

y

, we get

x = \sqrt{y}

. The radius of the disk at height

y

R(y) = \sqrt{y}

step 3

The limits of integration are from

y = 0

y = 4

because these are the bounds given for

y

step 4

Setting up the integral to find the volume:

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

step 5

Evaluating 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[ 8 - 0 \right] = 8\pi

Answer

The volume of the solid is

8\pi

cubic units.

Key Concept

Disk method for volume of solids of revolution

Explanation

The disk method involves integrating the area of circular disks along the axis of revolution to find the volume of the solid formed. In this case, the radius of each disk is given by the function

\sqrt{y}

, and the volume is found by integrating from the lower to the upper bound of

y