Question

Find the volume of the solid generated by revolving the region bounded by \(y=2 x-x^{2}\) and \(y=x\) about (a) the \(y\) -axis, (b) the line \(x=1\)

Short answer

The volume of the solid when revolved about the y-axis is \((2/5) \pi\) and when revolved around the line \(x=1\) is \(\pi/3\).

Step-by-step solution

Sketch the Region

Begin by graphing the functions \(y=2x-x^{2}\) and \(y=x\) on the same set of axes. Identify the region bounded by these two lines, which will be revolved about the y-axis and the line \(x=1\). You'll notice their intersection points are at (0,0) and (1,1).

Calculate the Volume using the Washer/Disc Method - Part (a)

If the region is revolved about the y-axis, we can calculate the volume using the washer method. Break the solid into tiny washers each with thickness \(dy\), outer radius \(\sqrt{y}\), and inner radius \(y\). The volume of each washer is the difference of the volumes of two cylinders, \(\pi*(\sqrt{y})^{2}*dy - \(\pi*(y)^{2}*dy\). We find the total volume by integrating the volume of each disk along the y-axis, from y=0 to y=1, \(V_{a}=\int_0^1 \[\pi(\sqrt{y})^{2}-\pi(y)^{2}\]\). Upon integration, we obtain \(V_a=(2/5) \pi\)

Calculate the Volume using the Washer/Disc Method - Part (b)

If the region is revolved about the line \(x=1\), the washer method has to be used again. The outer radius is now \(1-y\) and the inner radius is \(1-\sqrt{y}\). Thus, the volume of each disk is \(\pi*((1-y)^{2}*dy - (1-\sqrt{y})^{2}*dy)\). Integrating between y=0 and y=1, you obtain \(V_{b}=\int_0^1 \[\pi(1-y)^{2}-\pi(1-\sqrt{y})^2\]dy\). The result is \(V_b=\pi/3\).

Key concepts

Washer/Disc Method

The washer/disc method is a fundamental technique in calculus used for finding the volume of a solid of revolution. When a region in the xy-plane is revolved around an axis, the resulting solid can be approximated by stacking an infinite number of thin discs or washers.

Imagine slicing the solid perpendicular to the axis of revolution; each slice is a disc (if solid) or a washer (if hollow). The volume of each disc or washer can be calculated by \( \pi r^{2}dy \) or \( \pi (R^{2}-r^{2})dy \) respectively, where \( R \) and \( r \) are the radii of the outer and inner edges, and \( dy \) represents a tiny thickness of the washer.

By integrating this volume element from the start of the solid to the end, you can get the total volume of the solid of revolution. The choice of radii depends on the shape being revolved and the axis of revolution. Remember, integrating with respect to \( y \) implies horizontal 'washers', while integrating with respect to \( x \) implies vertical 'discs'.

Integration for Volume

Integration for volume is a powerful application of integral calculus. In the context of solids of revolution, the integral calculates the sum of the volumes of infinitely many washers or discs that make up the solid.

Let's break down the process into clear steps. First, express the radii of the discs/washers as functions of x or y, depending on whether you're integrating along the x-axis or y-axis. Next, set up the integral of the volume differential, which is essentially the volume formula of a disc or washer multiplied by \( dx \) or \( dy \) respectively, over the bounds of the solid.

For example, if we have the volume differential \( \pi (R^{2}-r^{2}) dy \) and we know the bounds for \( y \) are \( a \) to \( b \) , then our volume integral would be \( \int_{a}^{b} \pi (R(y)^{2}-r(y)^{2}) dy \) . Calculating this integral provides the volume of the entire solid. It's important to choose the correct bounds and accurately express the radii as functions, which often rely on the geometric interpretation of the problem at hand.

Solid Geometry Calculus

Solid geometry calculus is an exciting area of mathematics where calculus and geometry converge to solve problems about 3D shapes. Using calculus, especially integral calculus, we can derive general formulas to find volumes, surface areas, and more complex properties of various solids.

In the case of solids of revolution, the problem is usually how to compute the volume of a shape generated by rotating a two-dimensional region around an axis. The methods of washer/disc and cylindrical shells each have their place depending on the symmetry and axis of rotation of the solid. Considerations for setting up the correct integral involve identifying boundaries of the region, considering the axis of rotation, and determining the shape of the resulting solid.

Understanding the underlying geometry is key. For instance, when a function is rotated about the x-axis, the radius of a disc is typically the function value itself. However, if the rotation is about the y-axis or another line, geometry helps us in redefining these radii relative to the axis. Mastery of this concept not only helps in solving textbook problems but also in real-world applications such as engineering and physics where such calculations are crucial.