Question

FInd the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(x=\sqrt{y}+1, y=4, x=0, y=0 ;\) about the \(x\) -axis

Short answer

The volume is \(\frac{208\pi}{5}\).

Step-by-step solution

Sketch the Region R

The given curves are \(x = \sqrt{y} + 1\), \(y = 4\), \(x = 0\), and \(y = 0\). Start by sketching the parabola \(x = \sqrt{y} + 1\) which opens to the right. Next, draw the lines \(y = 4\) and \(y = 0\) as horizontal lines. The line \(x = 0\) is the y-axis. The region \(R\) is bounded by these curves and lies within the first quadrant, above the y-axis, and below \(y = 4\).

Show a Typical Slice

To find the volume using the shell method, consider a horizontal slice at a height \(y\). A typical slice is a rectangle with its height \(dy\) and its length equal to the distance between \(x = 0\) and \(x = \sqrt{y} + 1\), which is \(\sqrt{y} + 1\). This slice, when rotated around the x-axis, creates a cylindrical shell.

Approximate the Volume of the Shell

The formula for the volume of a cylindrical shell is \(2\pi \/ (\text{radius})\times (\text{height})\times (\text{thickness})\). In this case, the radius is \(y\), the height is \(\sqrt{y} + 1\), and the thickness is \(dy\). So, the approximate volume of the shell is \(2\pi y (\sqrt{y} + 1)dy\).

Set Up the Integral

We need to integrate from \(y = 0\) to \(y = 4\) to find the total volume. The integral is:\[V = \int_{0}^{4} 2\pi y (\sqrt{y} + 1) \, dy\].

Evaluate the Integral

First expand the integrand:\[2\pi \left( y\sqrt{y} + y \right) = 2\pi( y^{3/2} + y )\].Then integrate each term separately. Using the power rule, we get:\[V = 2\pi \left( \int_{0}^{4} y^{3/2} \, dy + \int_{0}^{4} y \, dy \right)\]\[= 2\pi \left( \frac{2}{5}y^{5/2} \Big|_0^4 + \frac{1}{2}y^2 \Big|_0^4 \right)\]\[= 2\pi \left( \frac{2}{5}(32) + 8 \right)\]\[= 2\pi \left( \frac{64}{5} + 8 \right)\]\[= 2\pi \left( \frac{64}{5} + \frac{40}{5} \right)\]\[= 2\pi \left( \frac{104}{5} \right)\]\[= \frac{208\pi}{5} \].

Key concepts

Shell Method

The Shell Method is a technique used in integral calculus for finding the volume of a solid of revolution. This method involves visualizing a solid as being made up of cylindrical shells, which are stacked together to fill the volume.
When you're working with the shell method, you first identify a typical slice or element of the area you're revolving. This slice is usually a rectangle whose height and base will define the dimensions of the shell once revolved. The shell's radius is the distance from the axis of rotation to the shell, the height is determined by the function, and the thickness is typically a small change in the variable of integration, like "dy" or "dx".
For example, in a situation where you're revolving around the x-axis, you would take a vertical slice and measure its width along the x direction. The formula for the volume of each shell is \( 2\pi \times \text{radius} \times \text{height} \times \text{thickness} \). This method is especially handy when the axis of rotation is parallel to the axis you are integrating along.

Integral Calculus

Integral calculus is a branch of mathematics that deals with integrals and their properties. In the context of finding the volume of solids, integral calculus is used to sum up infinitesimally small quantities to calculate a total.
It allows us to calculate the area under curves, which can be extended into volumes by revolving these areas around an axis. Integration is used to accumulate the volume of thin slices or shells to find the total volume of a solid.
When performing integration, you often convert a word problem or real-life scenario into a definite integral, which gives you a number that quantifies the concept being studied—in this case, volume. Whether you are using the disk/washer or shell method, integral calculus is your tool for turning continuous areas or lengths into quantitative measurements.

Revolving Around an Axis

Revolving around an axis is a central idea in calculating volumes of solids of revolution. Imagine a plane region being rotated about a straight line (the axis) to create a three-dimensional object.
The two typical methods to understand the volume of such objects are the disk/washer method and the shell method. The choice between these depends on the orientation of the rotation and whether it's simpler to integrate along the axis parallel or perpendicular to the axis.
For instance, if you're revolving around the x-axis, you may want to use the shell method if the function is easier to express with respect to y. This method essentially "wraps" the region around the axis, forming a solid with many cylindrical layers.

Definite Integration

Definite integration is the process of evaluating the area under a curve, and in the context of volumes of revolution, it determines the volume of a solid formed by revolving a region about an axis.
In definite integrals, limits are specified over which integration is performed. These limits define the bounds for the calculations, often corresponding to the minimum and maximum values that the function (or region) takes on in the problem.
In our problem, the definite integral is set from \(y=0\) to \(y=4\), corresponding to the bounding curves of the region being revolved. The integration then sums up the contributions from each infinitesimally small shell to compute the total volume. Calculating this integral provides you with an exact numerical value for the volume.