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

May 21, 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.

Generated Graph

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

step 2

The height of each cylindrical shell is given by

y = x^2

. The radius of each shell is

x

step 3

The volume of a thin shell is given by

2\pi \cdot \text{radius} \cdot \text{height} \cdot \text{thickness}

. Here, the thickness is

dx

step 4

The volume of the solid is obtained by integrating the volume of the thin shells from

x = 0

x = 2

step 5

Set up the integral:

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

step 6

Evaluate the integral:

2\pi \int_{0}^{2} x^3 \, dx = 2\pi \left[ \frac{x^4}{4} \right]_{0}^{2} = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right) = 2\pi \left( \frac{16}{4} \right) = 2\pi \cdot 4 = 8\pi

Answer

The volume of the solid is

8\pi

cubic units.

Key Concept

Cylindrical Shells Method

Explanation

The cylindrical shells method is used to find the volume of a solid of revolution by integrating the volume of thin cylindrical shells.