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

Nov 21, 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.

Here is a Step-by-Step solution: #step 1#

To find the volume of the solid obtained by rotating the region bounded by the curves around the y-axis, we will use the method of disks or washers. The volume

V

of the solid can be computed using the integral

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

where

R(y)

is the radius of the disk (or washer) at a given

y

-value, and

a

and

b

are the bounds of integration along the y-axis.

#step 2#

The region is bounded by

y = x^2

x = 0

, and

y = 4

. Since we are rotating around the y-axis, the radius of the disk at any point will be equal to the x-coordinate of the curve

y = x^2

. Solving for

x

, we get

x = \sqrt{y}

. The bounds of integration are from

y = 0

y = 4

, as given by the intersection points of the curves.

#step 3#

Setting up the integral to compute the volume, we have

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

#step 4#

Evaluating the integral, we get

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 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

Volume of a solid of revolution using the disk method

Explanation

The volume is found by integrating the area of circular disks along the axis of rotation, which in this case is the y-axis. The radius of each disk is determined by the x-value of the function

y = x^2

at a given y-value, and the integral is evaluated between the y-bounds of the region.