91Ó°ÊÓ

Coordinate axes are fundamental in creating a two-dimensional Cartesian plane. They consist of the x-axis, which runs horizontally, and the y-axis, running vertically. These axes allow us to identify any point in the plane using coordinates, written as \(x, y\).

• The x-coordinate indicates how far left or right the point is from the y-axis.
• The y-coordinate indicates how far up or down the point is from the x-axis.

In this exercise, we consider a point on the parabola \(y^2 = kx\) and define regions based on this person's projection onto the coordinate axes. This setup helps us calculate the areas of these regions and understand the subsequent revolve operations to find volumes.

The disk method is a technique used in calculus to find the volume of a solid of revolution. Imagine slicing the solid into thin disks perpendicular to the axis of revolution and calculating the volume of each disk. Adding up the disks' volumes gives us the total volume of the solid.For region A, revolving around the y-axis created a solid of revolution. We use the disk method to find its volume. For this specific case:\[ V_A = \pi \int_0^{y_0} \left( \frac{y^2}{k} \right)^2 dy = \pi \int_0^{y_0} \frac{y^4}{k^2} dy. \]

Steps to Solve using Disk Method:

• Identify the radius of each disk which is derived from the function of the curve you are revolving.
• Set up the integral across the limits that define the region.
• Evaluate the integral to find the exact volume.

This method works best when the solid’s cross-sectional area perpendicular to the axis of revolution can be easily determined and mathematically expressed.

The shell method is another integral calculus technique for finding volumes of solids of revolution. It revolves around using cylindrical shells and is particularly handy when integrating with respect to the opposite variable would be complicated, like when rotating about the y-axis.For region B, revolving about the y-axis, the shell method is applied. Here is how it plays out:\[ V_B = 2\pi \int_0^{x_0} x (y_0 - \sqrt{kx}) dx. \]

Steps in the Shell Method:

• Identify the height of the shell determined by the difference in function values within the region.
• Recognize the radius of the shell, equivalent to the distance from the axis of rotation.
• Set up and solve the integral over the defined bounds.

This method offers a flexible option for rotation around the y-axis, especially when a function is better viewed as parallel to the axis of rotation.

Integration is a mathematical operation used to sum quantities that vary continuously. It is integral (pun intended) in finding areas under curves and volumes of solids.In this exercise, integration helps calculate the volumes of the solids formed when regions A and B are revolved around an axis. When we speak of integrating in the context of volumes:

• It involves setting up an integral where each part of the solid is described by a small increment or slice.
• The integral bounds are determined by the region we are examining, such as \(0 \) to \(y_0\) for our parabola.
• Once the integral is set, it is evaluated to find exact numerical volumes.

Integration not only aids in understanding the spread of different areas but also consolidates how different volumes compare when solids are created by revolving them around coordinate axes.