Calculating the Volume of a Solid of Revolution Using Disk Method

Have you ever wondered how to find the volume of a solid formed by rotating a region around an axis? In this article, we will explore the process of using the disk method to find the volume of a solid of revolution. Specifically, we will delve into the example where we need to find the volume of the region bounded by y2x2, y4, and the y-axis for x≥0, when the region is rotated around the x-axis.

The Disk Method and the Volume Integral

To use the disk method, we first need to identify the bounds of integration. The curves yx2 and y4 intersect at x ±2. Since we are only considering x ≥ 0, the points of intersection are at (2,4) and (0,0).

Setting Up the Volume Integral

For the volume of the solid of revolution, we can write the integral as follows:

[V pi int_{a}^{b} [f(y)]^2 , dy]

where (f(y)) is the function that gives x in terms of y, which is x (sqrt{y}).

Evaluating the Integral

Now, substituting the relevant values and evaluating the integral:

[V pi int_{0}^{4} y , dy]

Carrying out the integration:

[V pi left[ frac{y^2}{2} right]_{0}^{4} pi left[ frac{4^2}{2} - frac{0^2}{2} right] pi left[ frac{16}{2} right] 8pi]

Therefore, the volume of the solid formed by rotating the region around the x-axis is (8pi).

Alternative Approach Using Strips and Centroids

Another approach to calculate the volume involves considering a strip parallel to the y-axis of width dx. The length of this strip is 4 - y, and the centroid of the strip is at a distance of y_s4 - y from the x-axis. The area of the strip is A_s4 - y. The volume can be expressed as the integral of the area times the distance from the centroid to the axis:

[V pi int_{0}^{2} (4 - y^2) , dy frac{128pi}{5}]

The final volume is approximately (80.42477) cubic units.

This example demonstrates the power of the disk method and how it can be applied to solve complex problems in calculus. Whether using the disk method or integrating strip areas, the key is to understand the bounds and the geometry involved in the problem.

Conclusion

Understanding how to find the volume of a solid of revolution using the disk method is essential for many applications in calculus and engineering. By mastering these techniques, you can solve a wide range of problems involving volumes and areas of revolution.