Problem 32 - Essential Calculus

Question

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\sqrt{2 x}, \quad y=x^{2} $$

Step-by-Step Solution

Answer

To find the volume of the solid, graph the functions, find the intersection points along the \(x\)-axis, apply the disk method of integration using the difference of the functions as the radial function, and integrate over the range defined by the intersection points.

Step 1: Plot the Functions

Enter the two functions \(y=\sqrt{2x}\) and \(y=x^{2}\) into a graphing utility to plot their graphs.

Step 2: Find Intersection Points

Find the points where the two functions intersect. This can be done by setting the two functions equal to one another as follows: \(\sqrt{2x} = x^{2}\)Solving this equation provides the intersection points along the \(x\)-axis, which will serve as the bounds of the integral.

Step 3: Apply Disk Method

Use the disk method formula for finding the volume of a solid of revolution, which is \( V = \pi \int_{a}^{b} [f(x)]^{2} dx\). The radial function \(f(x)\) is the distance from the surface of the solid to the axis of rotation at any given \(x\), which in this case is the difference between the two original functions, \(\sqrt{2x}\) and \(x^{2}\). So the formula becomes \( V = \pi \int_{a}^{b} [\sqrt{2x} - (x^{2})]^{2} dx\).

Step 4: Perform the Integration

Integrate the resulting function from the previous step over the interval \([a, b]\), which is obtained from the intersection points. This operation can be done with a graphing calculator or an online integration tool.

Key Concepts

Understanding the Disk Method

The disk method helps us find the volume of a solid formed by rotating a function around an axis. Imagine slicing the solid into thin disks, much like a stack of coins. Each disk's volume contributes to the total volume.

Here's how it works: if you have a function, say \( f(x) \), and you revolve it around the \( x \)-axis from \( a \) to \( b \), each disk has a radius \( f(x) \) and thickness \( dx \).

• The volume of each disk is \( \pi [f(x)]^2 dx \).
• Summing the volumes of these disks over the interval gives you the total volume.

The formula for the volume \( V \) becomes \( V = \pi \int_{a}^{b} [f(x)]^2 dx \).
This method works for any axis of rotation, just modify the function accordingly.

Using a Graphing Utility

Graphing utilities help visualize functions more easily. To use one, input your equations, like \( y = \sqrt{2x} \) and \( y = x^2 \), to see their curves.

Graphing utilities are great for:

• Finding where graphs intersect, which helps in determining bounds for integration.
• Confirming that the setup of your integral matches the rotation and bounds needed.

Many calculators have built-in capacity to handle integration, allowing you to compute volumes without manual calculations.

Determining Intersection Points

Intersection points mark where two functions meet or cross. In our context, they are essential in defining the limits of the integral.

To find them, set the functions equal: \( \sqrt{2x} = x^2 \).
Solve this equation to identify the \( x \)-values where these two curves intersect.

• These points serve as limits \( a \) and \( b \) when setting up the integral for disk method calculations.

Finding these points ensures that you only consider the region bounded by the functions when calculating the volume.

Integral Approximation Techniques

Integration helps us find areas under curves, and specifically here, it helps calculate volume.

While exact integration can be done manually, graphing utilities can approximate these integrals numerically.

• Numerical methods like Riemann sums or trapezoidal approximations serve well when dealing with more complicated functions.

For most practical purposes, these approximations are sufficiently accurate, especially when handled by technology.
Using a graphing utility, input the integral you've set up and compute the volume over the desired interval.