Problem 23 - Essential Calculus

Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=x^{2}+1, \quad y=-x^{2}+2 x+5, \quad x=0, \quad x=3 $$

Step-by-Step Solution

Answer

The volume of the solid figure of revolution is obtained by first graphing the equations to understand the bounded region, then setting up and evaluating the integral that represents the volume of the solid.

Step 1: Visualising and Understanding the Graphs and Bounded Region

The equations \(y=x^{2}+1\) and \(y=-x^{2}+2x+5\) should be graphed. Their intersection points, which become the limits of integration, should be identified. In addition, the bounded region between \(x=0\) and \(x=3\) should be recognized.

Step 2: Finding the Intersection Points

To find the points of intersection between the two curves, set the equations equal to each other: \(x^{2}+1=-x^{2}+2x+5\). Solving this will give the x values of the intersection points.

Step 3: Setting up the Volume Integral

Next, the formula for the volume of a solid of revolution using the method of disks is used, which is \(V = \pi \int_{a}^{b} [R(x)]^2 dx\), where \(R(x)\) is the radius of the disk at x. In this case, the outer radius, \(R(x)\), is defined by the equation \(y = -x^{2} + 2x + 5\) while the inner radius, \(r(x)\), is defined by \(y = x^{2} + 1\). Therefore, the integral becomes \(V = \pi \int_{a}^{b} [(R(x))^2 - (r(x))^2] dx\).

Step 4: Evaluating the Integral

Evaluate the integral over the defined limits to find the volume of the solid.

Key Concepts

Method of Disks

The Method of Disks is a technique used in integral calculus to determine the volume of a solid of revolution. Imagine a solid obtained by rotating a region around an axis. This method involves slicing the solid into thin, disk-shaped pieces, perpendicular to the axis of rotation.

Each disk has a small thickness, \(dx\), and a radius determined by the function or functions that define the boundary of the region. The volume of each disk is approximately \(\pi [R(x)]^2 dx\), where \(R(x)\) is the radius of the disk at a given point \((x)\). To find the total volume, integrate over the interval of interest.

This approach is particularly useful for solids with a clear axis of rotation and helps in visualizing the process by breaking down complex shapes into manageable parts.

Integral Calculus

Integral calculus deals with finding the accumulation of quantities and determines areas, volumes, and other concepts that arise from it. It involves the use of integrals to sum up an infinite number of infinitesimally small quantities.

In the context of finding volumes, the integral acts as a summation tool to add up the volumes of infinitely thin slices (like disks). The standard integral for volume involves the function for the area of a circle, \(\pi r^2\), integrated over a specified range.

Through integration, it is possible to calculate not just simple shapes but also more complex and irregular forms bounded by curves and lines.

Intersection Points

Intersection points are crucial when determining the limits for integration in volume problems. These points represent where two or more curves meet, and they help define the region to be revolved.

To find intersection points, set the equations of the curves equal and solve for the variable. In our example, we equate \(y = x^2 + 1\) and \(y = -x^2 + 2x + 5\) to find the x-values where the graphs intersect.

These x-values then become part of the integration limits, ensuring that the integral accounts for the full region bounded by these curves.

Graphing Equations

Graphing equations is a foundational step in solving volume of revolution problems. It helps visualize the area that will be revolved and provides insights into the behavior of the functions involved.

Begin by plotting each equation to see where they intersect and the overall shape of the region. By identifying features like intersection points and limits, you make the integration process more intuitive and grounded in a visual context.

Good graphing clarifies the problem and assists in setting up the correct integral, ensuring that all necessary components are considered for an accurate solution.