Problem 155 - Calculus Volume 1

Question

For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. [T] \(y=3 x^{3}-2, y=x,\) and \(x=2\) rotated around the \(y\) -axis.

Step-by-Step Solution

Answer

Volume is \(\frac{376\pi}{15}\).

Step 1: Sketch the Graph of the Region

Using graphing software, plot the functions \(y = 3x^3 - 2\), \(y = x\), and the vertical line \(x = 2\). Identify the region between these curves that will be rotated around the \(y\)-axis to form a solid.

Step 2: Set Up the Integral Using the Shell Method

Because the rotation is around the \(y\)-axis and the functions can be cumbersome when solved for \(x\), the shell method is more convenient. The formula for the volume \(V\) using the shell method is:\[ V = 2\pi \int_{a}^{b} x (f(x) - g(x)) \, dx \]where \(f(x)\) is the outer function and \(g(x)\) is the inner function for the given limits. Identify \(f(x) = 3x^3 - 2\) and \(g(x) = x\). The region of integration is from \(x = 0\) to \(x = 2\).

Step 3: Compute the Integral

Substitute \(f(x)\) and \(g(x)\) into the shell method formula:\[ V = 2\pi \int_{0}^{2} x ((3x^3 - 2) - x) \, dx \]Simplify and compute the integral:\[ V = 2\pi \int_{0}^{2} x (3x^3 - 2 - x) \, dx \]\[ = 2\pi \int_{0}^{2} x (3x^3 - x - 2) \, dx \]\[ = 2\pi \left[ \frac{3}{5}x^5 - \frac{1}{3}x^3 - x^2 \right]_{0}^{2} \]

Step 4: Evaluate the Integral

Evaluate the definite integral by substituting the upper and lower limits:\[ V = 2\pi \left( \left( \frac{3}{5}(2)^5 - \frac{1}{3}(2)^3 - (2)^2 \right) - \left( 0 \right) \right) \]Plug in the values and simplify:\[ = 2\pi (\frac{3}{5} \times 32 - \frac{1}{3} \times 8 - 4) \]\[ = 2\pi (\frac{96}{5} - \frac{8}{3} - 4) \]Find a common denominator and simplify:\[ = 2\pi (\frac{288}{15} - \frac{40}{15} - \frac{60}{15}) \]\[ = 2\pi (\frac{188}{15}) \]\[ = \frac{376\pi}{15} \]

Step 5: Conclusion: Compute Final Volume

The volume of the solid formed by rotating the region around the \(y\)-axis is \(\frac{376\pi}{15}\).

Key Concepts

Shell Method

The shell method is a technique for finding the volume of a solid of revolution. It is particularly useful when revolving around an axis, like in this exercise where the rotation is around the y-axis. This method uses cylindrical shells to approximate the volume. Each shell is a hollow cylinder, resembling a thin-walled tube.

The volume of such a shell can be obtained by considering:

• The height of the shell, which is the difference between the outer and the inner functions, denoted as \((f(x) - g(x))\).
• The circumference of the shell, calculated as \(2\pi x\) where \(x\) is the radius (distance from the axis of rotation).
• The thickness, an infinitesimally small value \(dx\).

The volume integral using the shell method can be expressed as: \[ V = 2\pi \, \int_{a}^{b} x (f(x) - g(x)) \, dx \] This method can simplify calculations when the rotation is around a vertical line such as the y-axis.

Definite Integral

Definite integrals are a fundamental concept used to calculate areas under curves and volumes of solids. In the context of volume of revolution, the definite integral helps in summing up infinitesimally small pieces to form a complete solid.

By using a definite integral from \(a\) to \(b\), we are essentially summing up all the 'small' contributions of the shell method over the interval, which gives the volume. This technique not only ensures precision but allows solving problems where calculating manually would be complex.

The limits \(a\) and \(b\) are determined based on the interval along the x-axis, representing the span across which the region is being evaluated. When solved, a definite integral provides an exact number representing the total volume.

Rotating Around an Axis

In calculus, obtaining the volume of a solid by rotating a region around an axis is a common application. This involves revolving a region, defined by functions and limits, around an axis to sweep out a three-dimensional object.

Depending on the axis of rotation (x-axis, y-axis, or another specified line), different integration methods are used, such as the disk, washer, or shell method. In the shell method, since rotation is around the y-axis, shells are formed parallel to the x-axis.

Solving these problems involves:

• Identifying the region to be revolved.
• Choosing the correct method (such as the shell method) based on the axis and setup.
• Setting up the appropriate integral to compute the volume.

Understanding the axis of rotation is key to determining how the solid is generated and, hence, how to apply the correct mathematical tools.

Graphing Technology in Calculus

Graphing technology plays an invaluable role in calculus, especially when visualizing functions and regions for complex problems. Tools like graphing calculators and computer software help with:

• Plotting functions accurately to identify important features such as intersections, regions of interest, and axes.
• Determining regions to be revolved, making it easier to visualize the problem.
• Cross-referencing calculations with visual outputs for verification.

For students, using such technology enhances understanding by allowing them to visually see the effects of rotation and the shape of the resulting solid, which are vital for grasping concepts of the volume of revolution.

Moreover, technology helps verify solutions, offering a practical way to explore and comprehend calculus concepts deeply.