Problem 20 - The Calculus with Analytic Geometry

Question

The region bounded by a pentagon having vertices at \((-4,4),(-2,0),(0,8),(2,0)\), and \((4,4)\) is revolved about the \(x\) axis. Find the volume of the solid generated.

Step-by-Step Solution

Answer

The volume is found by summing integrals of smaller regions, using the disk method.

Step 1: Understand the Shape

Identify the pentagon's vertices, which are \((-4,4),(-2,0),(0,8),(2,0), \ \text{and} (4,4)\). Plot these points on a coordinate system to visualize the pentagon.

Step 2: Divide the Pentagon

Break down the pentagon into simpler geometric shapes, such as triangles. For clarity and ease of calculation, consider dividing it into triangles and trapezoids.

Step 3: Use the Disk Method

When the region is revolved around the \(x\)-axis, you can use the disk method (or washer method if there are hollow sections) to find the volume of the solid. Each triangular or trapezoidal section can be analyzed separately, and the volume contributions can be summed.

Step 4: Set Up Integrals

Set up the integrals corresponding to each section being revolved. For instance, consider sections between \(x = -4\) to \ (x = -2)\ for one triangular part, and similarly, for the other sections. The volume of the solid formed by rotating a region bounded by \(y = f(x)\) and \(y = g(x)\) from \a\ to \b\ about the \(x\)-axis can be expressed as: \ V = \pi \int_{a}^{b} [f(x)^2 - g(x)^2] dx \.

Step 5: Evaluate the Integrals

Evaluate each integral. For example: For the section from \ x = -4 \ to \ x = -2\, integrate the radius squared (in terms of \(x\) and \y\ values from the vertices) about \( x \). Repeat similarly for other sections filling the pentagon, using sum of integrals to obtain total volume.

Step 6: Compute the Total Volume

Sum all evaluated integrals obtained from previous step. This gives the total volume of the solid generated.

Key Concepts

disk method

The disk method is a way to find the volume of a solid of revolution. Imagine a shape that rotates around an axis to form a three-dimensional object. By slicing this object into infinitely thin disks, we can compute its volume. Each disk has a radius equal to the function value at that point and a very small thickness. The volume of each infinitesimal disk is \(\text{dV} = \pi [f(x)]^2 \text{d}x\).

When summed (integrated) from one boundary to another, these disks give the total volume of the solid. This integral is expressed as:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \text{d}x \]

For the given exercise, the pentagon is divided into simpler shapes like triangles and trapezoids. You apply the disk method to each section formed by these shapes.

integral calculus

Integral calculus is a branch of mathematics focused on finding the total size, length, area, or volume of an object. It is the process of summing infinitesimally small quantities. Bar Integral calculus, you can find the area under a curve, the total accumulated change, or the volume of an irregular shape.

In this exercise, we use integral calculus to find the volume of a solid of revolution. The pentagon's shaped, divided up and rotated around the x-axis, gives a complex 3D figure.

By setting up integrals that represent the volume of each section individually and then evaluating them, you sum these integrals to find the whole volume of the solid.

geometric decomposition

Geometric decomposition is the process of breaking down a complex shape into simpler, more manageable parts. This method helps making calculations on the volumes, areas, or other properties easier.

In our case, we divide the pentagon into shapes such as triangles and trapezoids. These basic geometric shapes have well-defined formulas that make it feasible to set up integrals.

For example, the pentagon in the problem can be split into a few triangles and trapezoids. Each of these shapes has boundaries defined by the x-coordinates and y-coordinates of the pentagon's vertices. This makes setting up integrals much simpler and allows for the application of the disk method.

coordinate geometry

Coordinate geometry, or analytic geometry, uses coordinate systems to solve geometric problems. By plotting points, lines, and curves on the coordinate plane, it becomes easier to visualize and solve complex geometry tasks.

The exercise involves plotting the five vertices of the pentagon: \((-4,4), (-2,0), (0,8), (2,0), (4,4)\). Once plotted, you can see how the pentagon appears and identify sections to split it into simpler shapes.

Using coordinate geometry, you determine the function values—essentially the heights of the slices of the solid of revolution. These are the values plugged into your integrals. Coordinate geometry thus lays the foundation for setting up the problem and solving it using methods like integral calculus and the disk method.