Calculus Challenge: Volume of Solids of Revolution

The problem at hand is a classic application of calculus: finding the volume of a solid generated by rotating a region in the first quadrant formed by the intersection of the parabola y x^2 - 3x and the line y 5x. This article explores the different methods to calculate the volume, focusing on rotation about the x-axis and y-axis.

Rotation about the y-axis

The first step is to determine the points of intersection between the parabola and the line. By setting y x^2 - 3x equal to y 5x, we get:

x^2 - 3x 5x, which simplifies to:

x^2 - 8x 0, giving us roots at x 0 and x 8.

The region of interest is within the first quadrant, so we focus on the interval from x 0 to x 8. To find the volume of the solid formed by rotating this region about the y-axis, we can use the method of cylindrical shells. The formula for the volume is given by:

V 2π∫_{a}^{b} x (f(x) - g(x)) dx

where f(x) 5x and g(x) x^2 - 3x.

Substituting the functions, we get:

V 2π∫_{0}^{8} x (5x - (x^2 - 3x)) dx 2π∫_{0}^{8} (25x - x^3) dx

Evaluating the integral:

V 2π [12.5x^2 - 0.25x^4]_0^8 2π [100 - 4096/16] 2π (100 - 256) -456π

The negative sign indicates the direction of rotation; the correct volume is:

V 225π or approximately 706.86 cubic units.

Rotation about the x-axis

Next, we explore the rotation of the same region about the x-axis, using the washer method. The volume of the solid generated is given by:

V π∫_{a}^{b} (f(x)^2 - g(x)^2) dx

Here, f(x) 5x and g(x) x^2 - 3x. Substituting these into the integral:

V π∫_{0}^{8} (25x^2 - (x^2 - 3x)^2) dx π∫_{0}^{8} (25x^2 - x^4 6x^3 - 9x^2) dx

Simplifying the integrand:

V π∫_{0}^{8} (-x^4 6x^3 16x^2) dx

Evaluating the integral:

V π [ -0.2x^5 1.5x^4 5.33x^3]_0^8 π ( -243/5 1296/2 1728/3) 2329.167 π

The volume of the solid generated by rotating the region about the x-axis is approximately:

V 7317.29 cubic units.

Conclusion

The volume of the solid formed by rotating the region enclosed by the curves y x^2 - 3x and y 5x about different axes can be calculated using the appropriate methods (cylindrical shells for y-axis and washers for x-axis). The values obtained demonstrate the importance of choosing the correct method based on the axis of rotation and the given functions.

Key Takeaways:

Knowing the points of intersection between the parabola and the line is crucial for setting up the integration limits. The method used (washes for x-axis, cylindrical shells for y-axis) significantly affects the integral setup and the final volume calculation. Understanding the geometry of the solid formed by rotation helps in visualizing and solving the problem accurately.