Volume of Solid of Revolution: Rotating Around the Y-Axis

Objective

• Find the volume of a solid of revolution by rotating a given curve around the y-axis, between y = 1 and y = 4.

Key Concepts

• Solid of Revolution: A three-dimensional object obtained by rotating a two-dimensional area around an axis.
• Curve: The curve under consideration is (y = x^2).
• Rotation Axis Change: Unlike previous examples where rotation was around the x-axis, this example rotates the area around the y-axis.

Visualization

• The shape formed looks like an upside-down truffle.
• The structure is visualized by rotating the curve of (y = x^2) around the y-axis.
• The visualization involves imagining discs with a thickness of (dy).

Calculating the Volume

1. Setup of Integral with respect to y:

   • Equation Change: Solve (y = x^2) for x, resulting in (x = \sqrt{y}).
   • Radius of Discs: At a given y, the radius of the disc is (\sqrt{y}).

2. Area of Disc:

   • Formula for area: (\pi r^2)
   • Substituting the radius: Area = (\pi (\sqrt{y})^2 = \pi y)

3. Volume of Each Disc:

   • Volume = Area (\times) thickness = (\pi y \times dy)

4. Definite Integral:

   • Integrate from (y = 1) to (y = 4) to find total volume:
   • [ \text{Volume} = \int_{1}^{4} \pi y, dy ]
   • Taking (\pi) out of the integral: (\pi \int_{1}^{4} y, dy)

5. Evaluation of Integral:

   • Antiderivative of (y) is (\frac{y^2}{2})
   • Evaluate from 1 to 4: [ \pi \left( \frac{4^2}{2} - \frac{1^2}{2} \right) = \pi \left( \frac{16}{2} - \frac{1}{2} \right) = \pi \left( 8 - \frac{1}{2} \right) = \pi \cdot \frac{15}{2} ]

Conclusion

• The volume of the solid formed by rotating the curve (y = x^2) from (y = 1) to (y = 4) around the y-axis is (\frac{15}{2} \pi) or equivalently (7.5\pi).