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

Oct 11, 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

To find 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 method of cylindrical shells. The volume

V

is given by the integral:

V = 2\pi \int_{0}^{2} x(4 - x^2) \, dx

step 2

We first determine the limits of integration. The curve

y = x^2

intersects

y = 4

x = 2

. Thus, we integrate from

0

2

step 3

Now we compute the integral:

V = 2\pi \int_{0}^{2} (4x - x^3) \, dx

step 4

Evaluating the integral, we have:

V = 2\pi \left[ 2x^2 - \frac{x^4}{4} \right]_{0}^{2} = 2\pi \left[ 8 - 4 \right] = 8\pi

Answer

The volume of the solid is

8\pi

cubic units.

Key Concept

The method of cylindrical shells is used to find the volume of solids of revolution.

Explanation

By integrating the product of the radius and height of the shells, we can calculate the total volume of the solid formed by rotation.