Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(y\) -axis. $$ y=3(2-x), \quad y=0, \quad x=0 $$

Short answer

The volume of the solid generated by revolving the given region around the y-axis is \(4 \pi\) cubic units.

Step-by-step solution

Write Equation for Radius

The radius is the distance between \(x = 0\) (the axis of rotation) and \(x = 2 - y / 3\). Since the solid is obtained by rotating with respect to the y-axis, the radius is \(r = x\). In terms of \(y\), \(r = 2 - y / 3\).

Write the Integral for Volume

The volume \(V\) of the solid obtained by rotating the graph around the y-axis from \(y=a\) to \(y=b\) is given by the integral from \(a\) to \(b\) of the cross-sectional area of the disc, which is \(\pi r^2\), along the y-axis. Therefore, \(V = \int_a^b \pi (r(y))^2 dy\), where \(r(y) = 2 - y / 3\), for \(y\) ranging from 0 to 6.

Compute the Integral

Substitute \(r(y)\) into the integral and compute: \[V = \int_0^6 \pi (2 - y/3)^2 dy\]. Using power rule and constant rule in integral calculus, expand the square and then integrate term by term. Here you will require a bit of algebraic simplification before you can easily compute the integral.

Evaluate the Definite Integral

After performing the integration process, the resulting answer should be evaluated at the definite integral boundaries which are 0 and 6. Subtraction should then be performed based on these evaluated results.

Key concepts

Disk Method

Understanding the disk method is essential to finding the volume of solids formed by rotation. Imagine slicing the solid into thin, circular discs perpendicular to the axis of rotation. Each disk's volume can be thought of as the volume of a cylinder with a very small height: \( V_{\text{disk}} = \pi r^2 h \), where \( r \) is the radius of the disk, and \( h \) is its height. In calculus, when we want to find the total volume of the solid, we sum up these infinitesimally small volumes using an integral. In our specific problem, the solid is generated by rotating a region bounded by certain graphs, and the radius of each disk is a function of \( y \) because we're rotating about the \( y \) -axis. So, we use the disk method to write down an integral that will give us the total volume once evaluated.

Definite Integral

A definite integral in calculus represents the signed area under a curve between two points on the horizontal axis. It is expressed as \( \int_a^b f(x) \, dx \), where \( f(x) \) is a function, and \( a \) and \( b \) are the bounds of integration. When calculating the volume of a solid of revolution, we use the definite integral to add up the infinite number of circular disk volumes as we move along the axis of rotation. The limits of integration, in this case, are the \( y \) values that bound the region being revolved. Evaluating the integral means finding the total volume by subtracting the volume calculated at the lower bound from the volume calculated at the upper bound.

Calculus Application

Calculus is a powerful tool for understanding and solving problems involving change and motion, and one such application is in computing the volume of solids of revolution. By applying integral calculus to the function that defines the radius of the disks, we are essentially summing an infinite number of cross-sectional areas to find the total volume. This application necessitates a blend of geometric insight—understanding the shape and orientation of the solid—and algebraic skill, to manipulate and integrate the function that represents the solid's varying radius.

Rotational Volume

Rotational volume refers to the volume of a three-dimensional object that is created by revolving a two-dimensional region around a line (the axis of rotation). The region could be between curves, lines, or both. In the problem provided, we're interested in the area under the line \( y = 3(2 - x) \) and above the \( x \) axis (since \( y = 0 \) indicates the \( x \) axis), rotated around the \( y \) axis. Calculating the rotational volume involves finding the radius of rotation at different points, then integrating across the region to sum all the disk volumes as it sweeps through the 360-degree rotation.