Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. \(y=x^{2}, \quad y=4 x-x^{2}\) (a) the \(x\) -axis (b) the line \(y=6\)

Short answer

The volume of the solid obtained by revolving the given region around the x-axis is the result of the integral in step 2, and the one from revolving around the line \(y = 6\) comes from the integral in step 3.

Step-by-step solution

Find the Points of Intersection

The points of intersection are the solutions to the equation \(x^2 = 4x - x^2\). Adding \(x^2\) to both sides gives \(2x^2 = 4x\), or \(x^2 = 2x\). This can be written as \(x(x - 2) = 0\), which means \(x = 0\) or \(x = 2\). So, the region is bounded by the x-values 0 and 2.

Volume Revolved Around the x-axis (a)

The volume of the solid formed by revolving the region around the x-axis is given by \(\int_a^b (R(x)^2 - r(x)^2) dx\), where R(x) is the outer radius and r(x) is the inner radius. Here, \(R(x) = 4x - x^2\) and \(r(x) = x^2\). Thus, the volume V is \(\int_0^2 ((4x - x^2)^2 - (x^2)^2) dx\). Computing this integral gives the desired volume.

Volume Revolved Around the Line y=6 (b)

When revolving the region around the line \(y = 6\), the shell method is employed. The volume is given by \(\int_a^b 2π(p(x)h(x)) dx\) where \(p(x)\) is the distance of the shell to the rotation axis and \(h(x)\) is the height of the shell. The height is the difference between the curves, \(h(x) = 4x - x^2 - x^2\) and the distance to the axis of rotation is 6 minus the upper function, \(p(x) = 6 - (4x - x^2)\). The volume V is \(\int_0^2 2π((6 - (4x - x^2))(4x - 2x^2)) dx\). The computation of this integral delivers the volume.

Key concepts

Methods of Finding Volume

Finding the volume of a solid formed by revolving a region around a particular axis is a classic problem in calculus. There are several methods to achieve this, but two particularly important ones are the disc method and the shell method. The choice between these methods typically depends on the shape of the solid and the axis of revolution.

The disc method is often used when the solid can be described as a stack of discs (or washers) whose radii vary along a single axis. This method is straightforward for solids of revolution around the x-axis or y-axis since each cross-section perpendicular to the axis of rotation forms a disc. On the other hand, the shell method is advantageous when dealing with solids that can be seen as cylindrical shells. This method is particularly useful when revolving around a vertical or horizontal line that isn't an axis.

Integrals in Calculus

In calculus, integrals are used to calculate many things, one of which is the area under a curve, a central concept that extends to finding volumes. Calculating the volume of a solid of revolution involves taking an infinite sum of infinitely thin slices (discs or shells) that make up the solid, a process known as integration.

When the region is revolved around an axis, the integral calculates the sum of all the small slices of volume. For this reason, students must understand how to set up an integral to reflect the solid's geometry. Integrals can represent accumulation of quantities; in the case of volume, the integral accumulates the volume of each infinitesimally thin disc or shell across the interval of integration.

Shell Method

The shell method is particularly useful when the solid of revolution is generated around a line that isn't one of the coordinate axes. This method visualizes the solid as being composed of concentric cylindrical shells.

The volume of each individual shell can be considered as the surface area of the cylinder times its thickness. To find the volume, we integrate the volume of these shells over the interval of interest. The formula used is \( V = \int_a^b 2\pi (p(x)h(x)) dx \) where \( p(x) \) represents the distance to the axis of revolution (the radius of the shell), and \( h(x) \) is the height of the shell. The product of \( 2\pi \) and the integration represents summing up the circumferences of the cylindrical shells times their respective heights and thicknesses to obtain the total volume.

Disc Method

The disc method is a straightforward approach to finding the volume of a solid of revolution when the cross-sections perpendicular to the axis of rotation form discs. Each disc's thickness is infinitesimally small, and its radius is determined by the value of the function at a given point.

For volumes of solids of revolution around the x-axis, the formula used is \( V = \int_a^b (R(x)^2 - r(x)^2) dx \) where \( R(x) \) is the outer radius function and \( r(x) \) is the inner radius function (used in the case of washers, where there is an empty space in the middle). The integrand, \( (R(x)^2 - r(x)^2) \) represents the area of one disc or washer, and integrating this area over the interval from \( a \) to \( b \) helps us find the volume of the entire solid.