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. \(x=\sqrt{2 y}+1, y=2, x=0, y=0 ;\) about the line \(y=3\)

Short answer

The volume of the solid is \(2\pi(16\sqrt{2} + \frac{32}{5})\).

Step-by-step solution

Sketch the Region R

First, let's identify the boundaries of the region. The curve given is \(x=\sqrt{2y}+1\). The region \(R\) is also bounded by the lines \(y=2\), \(x=0\), and \(y=0\). Sketch these on the coordinate plane. This region is situated in the first quadrant and has a curved upper boundary from the graph of \(x=\sqrt{2y}+1\) and a straight horizontal line at \(y=2\). It is vertical between \(y=0\) and \(y=2\) under the curve of \(x=\sqrt{2y}+1\).

Show a Typical Rectangular Slice

Take a typical slice parallel to the y-axis within the region. This slice is positioned at a point \(y\) where \(0\leq y \leq 2\). This slice has a small thickness \(\Delta y\) and extends horizontally from \(x=0\) to \(x=\sqrt{2y}+1\). This rectangle will be used to form cylindrical shells when revolved around the line \(y=3\).

Write Formula for Volume of Shell

For the shell method, the volume \(dV\) of a cylindrical shell is given by \(dV = 2\pi (\text{radius})(\text{height})(\text{thickness})\). Here, the radius is \(3-y\) (the distance from the line \(y=3\)), the height is \(\sqrt{2y}+1\), and the thickness is \(\Delta y\). So, \(dV = 2\pi (3-y)(\sqrt{2y}+1) \Delta y\).

Set Up the Integral

To find the total volume, integrate the expression for \(dV\) from \(y=0\) to \(y=2\). The integral becomes: \[V = \int_{0}^{2} 2\pi (3-y)(\sqrt{2y}+1) \, dy\]

Evaluate the Integral

To evaluate the integral, distribute and simplify the expression:\[V = 2\pi \int_{0}^{2} ((3-y)(\sqrt{2y}+1)) \, dy = 2\pi \int_{0}^{2} (3\sqrt{2y} + 3 - y\sqrt{2y} - y)\, dy\]Break it into simpler integrals:\[V = 2\pi \left( \int_{0}^{2} 3\sqrt{2y} \, dy + \int_{0}^{2} 3 \, dy - \int_{0}^{2} y\sqrt{2y} \, dy - \int_{0}^{2} y \, dy \right)\]Compute each integral:1. \(\int 3\sqrt{2y} \, dy = \int 3(2)^{1/2}y^{1/2} \, dy = 6(2)^{1/2} \cdot \frac{2}{3}y^{3/2} \big|_0^2 = 4\sqrt{2}\cdot 2\)2. \(\int 3 \, dy = 3y \big|_0^2 = 6\)3. \(\int y\sqrt{2y} \, dy = \int \sqrt{2}(y^{3/2}) \, dy = \frac{2\sqrt{2}}{5}y^{5/2} \big|_0^2 = \frac{2\sqrt{2}}{5}\cdot 4\sqrt{2}\)4. \(\int y \, dy = \frac{1}{2}y^2 \big|_0^2 = 2\)Combine these results:\[V = 2\pi \left( 16\sqrt{2} + 6 - \frac{8}{5}\cdot 2 - 2 \right) = 2\pi \left( 16\sqrt{2} + 4 - \frac{16}{5} \right) = 2\pi \left( 16\sqrt{2} + \frac{20}{5} - \frac{16}{5} \right)\]Simplify further and find the exact volume.

Key concepts

Cylindrical Shell Method

The cylindrical shell method is a technique in calculus used to find the volume of a solid of revolution. Imagine taking a small rectangle from the region you're interested in and rotating that around a line. This creates a shell, somewhat like a hollow cylinder. By stacking all these shells together, you can approximate and eventually calculate the total volume of the solid. Here's how it works:- You begin by choosing a vertical or horizontal slice of the region, depending on which provides an easier integration.- Each slice's rotation around the axis of revolution forms a cylindrical shell.- The volume of each shell is calculated using the formula: \(dV = 2\pi (\text{radius})(\text{height})(\text{thickness})\).With this method, it’s vital to understand the geometric relationship between the slice and the axis of rotation, as the distance from the axis affects the radius of the shell. The examples you’ve practiced help build intuition about these relationships and set up the correct integral easily.

Integration

Integration is the process of finding the whole from its parts. In the context of the cylindrical shell method, integration is used to sum up the infinitesimally small volumes of the cylindrical shells to find the entire volume of the solid of revolution. Here's what you need to know:- **Setting Up the Integral**: For the slices, calculate the volume element \(dV\). These little volumes are functions of \(y\) or \(x\), depending on the method and specific problem you're solving.- **The Integration Limits**: These depend on the range over which the function and the region are defined. For the shell method, these are often the bounds for which the slice slices through the region.- **Solving the Integral**: Break down complex expressions into simple integrals. Use basic antiderivatives and fundamental integration rules to solve them. Integration not only combines small pieces into a whole but demands a good grasp of function manipulation and algebra.

Solid of Revolution

A solid of revolution is a 3D object obtained by rotating a 2D region around an axis. In practical terms, imagine taking a curve or shape on a piece of paper and turning it around a straight line (axis). This rotation forms a solid shape. Key Points: - **Axis of Rotation**: Determines how the shape is revolved and thus changes the resulting solid. - **Boundaries**: The edges of the 2D area are important as they define the limits of rotation. Without clearly set boundaries, it’s hard to define the 3D shape properly. - **Visualizing**: Using accurate sketches and imagining the rotation can greatly assist in setting up the problem correctly. Understanding solids of revolution is essential in real-world applications like engineering, where modeling rounded solids arise frequently. They give a tangible way to apply calculus concepts as real-world problems often require these methods to find solutions.

Calculus Problems

Calculus problems involving finding volumes are common exercises for developing both analytical skills and understanding of real-world applications of mathematics. They may seem challenging at first, but they develop important skills in thinking about how shapes and regions behave under transformations like rotations. Tips for Tackling These Problems: - **Start with a Sketch**: Always visualize the region and the solid. A good sketch provides clarity on what is being revolved and how the resulting shape looks. - **Identify Bounds and Limits**: Before jumping into integrations, establish the correct limits. This ensures that you are only considering the region of interest. - **Break Down Complexes**: If the integrand looks tough, simplify it into parts. It’s easier to tackle multiple small integrals than one complicated one. - **Check Dimensions and Units**: Always ensure that your final answer makes sense dimensionally. Volume should be in cubic units. With practice and perseverance, these calculus problems become a powerful tool in your academic toolkit, ready for use in both theoretical and practical settings.