Volume of the Solid Generated by Rotating a Region about a Vertical Line

Rotation problems in calculus often involve calculating the volume of a solid formed by revolving a region around a specified axis. In this article, we'll explore the volume of the solid generated by revolving the region bounded by the curve y^2 16x and the line x 4 around the line x 4. This problem involves both analysis and application of calculus concepts, making it a valuable exercise for students and professionals in engineering, physics, and mathematics.

Understanding the Region and the Problem

The curve given is y^2 16x, which can be described by a circle in the x-y plane when considering its geometric representation. The line x 4 is a vertical line that intersects the curve at the points where x 4, leading to y ±8. These points of intersection A(4, 8) and B(4, -8) define the bounded region AOB.

Calculation of the Volume

To find the volume of the solid generated by revolving the region AOB around the line x 4, we can use the Method of Cylindrical Shells.

Using the Method of Cylindrical Shells

The volume V can be calculated using the formula for the Method of Cylindrical Shells:

V 2π ∫ from -8 to 8 4 - x dx

where x is the distance from the axis of rotation x 4, and y is the variable of integration. However, since the region is symmetric, we can also utilize the Disk Method for better simplification:

V π ∫ from -8 to 8 (4 - x2) dy

This simplifies the calculation as shown:

V π ∫ from -8 to 8 (4 - y2/16) dy Let's integrate this function step by step:

V π [4y - y5/1280] from -8 to 8 Evaluating at the limits:

V π [4(8) - (85/1280) - (4(-8) - (-85/1280))] V π [32 - (32/5) 32 - (32/5)]

Simplifying, we get:

V (64π - 64π/5) (320π - 64π)/5 256π/5

Therefore, the volume is:

V 2048π/15 cubic units.

Alternative Method: Cylindrical Shells

Alternatively, we can use the Method of Cylindrical Shells directly by integrating:

V 2π ∫ from 0 to 4 4 - √x (4 - √x) dx

This simplifies to:

V 2π ∫ from 0 to 4 (16 - 8√x - x) dx

Integrating, we get:

V 2π [16x - 8(2/3)x3/2 - (1/2)x2] from 0 to 4

Evaluating at the limits:

V 2π [64 - (128/3) - 8] 2π [48 - 128/3] 2π [384/3 - 128/3] 2π(256/3)

Therefore, the volume is:

V 2048π/15 cubic units.

Conclusion

The problem of finding the volume of the solid generated by revolving the region bounded by the curve y^2 16x and the line x 4 around the line x 4 is an excellent exercise in calculus. It demonstrates the application of cylindrical shells and disk methods and provides insight into the importance of symmetry in simplifying integrals. The solution yields a volume of 2048π/15 cubic units, or approximately 428.932 cubic units.