AskSia

Plus

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

Jan 3, 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.

Solution by Steps

step 1

Identify the method of disks/washers to compute the volume of the solid of revolution

step 2

Set up the integral for the volume using the washer method, where the outer radius

R(y)

\sqrt{y}

and the inner radius

r(y)

is 0, since the region is rotated about the y-axis

step 3

Write the volume integral:

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

step 4

Simplify the integral:

V = \pi \int_{0}^{4} y dy

step 5

Compute the integral:

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

step 6

Evaluate the definite integral:

V = \pi \left( \frac{4^2}{2} - \frac{0^2}{2} \right)

step 7

Simplify to find the volume:

V = \pi \left( \frac{16}{2} \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 washer method

Explanation

The washer method involves integrating the area of a washer, which is the region between two concentric circles, as it revolves around an axis. In this case, the outer radius is the square root of y, and there is no inner radius since the solid is rotated around the y-axis.