Question

Sketch the region \(R\) bounded by the graphs of the given equations, and show a typical vertical slice. Then find the volume of the solid generated by revolving \(R\) about the \(x\) -axis. \(y=x^{2 / 3}, y=0\), between \(x=1\) and \(x=27\)

Short answer

The volume is \( \frac{6558\pi}{7} \) cubic units.

Step-by-step solution

Sketch the Region

First, we need to understand the region we are dealing with. The curve \( y = x^{2/3} \) starts from the origin \((0,0)\) and increases as \(x\) increases. The other boundary is the line \( y = 0 \), which is the x-axis. We are interested in the part between \( x = 1 \) and \( x = 27 \). The region is a strip above the x-axis under the curve \( y = x^{2/3} \).

Set up the Integral for Volume

To find the volume of the solid generated by revolving this region around the x-axis, we use the disk method. The formula for the volume of revolution using the disk method is \( V = \pi \int_a^b [f(x)]^2 \, dx \), where \( f(x) \) is the height of each disk. Here, \( f(x) = x^{2/3} \), \( a = 1 \), and \( b = 27 \).

Evaluate the Integral

Substitute the function into the integral: \[V = \pi \int_1^{27} (x^{2/3})^2 \, dx = \pi \int_1^{27} x^{4/3} \, dx\]To solve the integral, find the antiderivative of \( x^{4/3} \). Using the power rule, the antiderivative of \( x^{4/3} \) is \( \frac{3}{7}x^{7/3} \). Evaluate this from 1 to 27.

Calculate the Definite Integral

Plug in the bounds into the antiderivative: \[V = \pi \left[ \frac{3}{7}x^{7/3} \right]_1^{27} = \pi \left( \frac{3}{7} \times 27^{7/3} - \frac{3}{7} \times 1^{7/3} \right)\]Since \( 27^{7/3} = (3^3)^{7/3} = 3^7 = 2187 \), we substitute it into the equation:

Simplify and Compute the Volume

Simplify the expression: \[V = \pi \left( \frac{3}{7} \times 2187 - \frac{3}{7} \times 1 \right) = \pi \left( \frac{6561}{7} - \frac{3}{7} \right)\]This simplifies to: \[V = \pi \times \frac{6558}{7}\] Calculate to find \[V = \frac{6558\pi}{7}\] cubic units.

Key concepts

Disk Method

The Disk Method is a powerful technique in calculus used to find the volume of solids of revolution. This method involves revolving a region around an axis to form a three-dimensional solid. It's particularly useful when the cross-sections of the solid perpendicular to the axis of revolution are disks.

• The formula to find the volume using the Disk Method is: \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]where \( f(x) \) is the function defining the upper boundary of the region, and \( a \) and \( b \) are the limits of integration.
• Each disk has a radius of \( f(x) \), which is the distance from the x-axis to the curve.
• The height of each small disk is an infinitesimal change in \( x \), denoted as \( dx \), which gives the thickness of the disks.

In our problem, we're revolving the region bounded by \( y = x^{2/3} \) and \( y = 0 \) about the x-axis from \( x = 1 \) to \( x = 27 \). This means the volume has to be calculated by integrating this function squared over the given interval.

Definite Integral

A Definite Integral is a key mathematical concept used to calculate the "net area" between a function and the x-axis over a specific interval. It extends beyond finding areas; in context, it's critical for determining volumes, especially in the Disk Method.

• Its proper notation is \( \int_{a}^{b} f(x) \, dx \), where \([a, b]\) is the closed interval over which we integrate.
• In the Disk Method, the definite integral sums up the infinitesimally small volumes of the disks from \( a \) to \( b \).
• The result gives the accumulated volume of the solid formed by revolution.

For our exercise, we compute the definite integral of \( (x^{2/3})^2 \) from 1 to 27, summing up all the infinitesimally small disk volumes between these bounds to find the total volume.

Antiderivatives

Antiderivatives, also known as indefinite integrals, are fundamental when evaluating definite integrals. Finding an antiderivative means looking for a function whose derivative is the given function.

• The process involves using rules such as the power rule, which states that the antiderivative of \( x^{n} \) is \( \frac{x^{n+1}}{n+1} \), where \( n eq -1 \).
• In our exercise, the antiderivative of \( x^{4/3} \) is derived using the power rule, resulting in \( \frac{3}{7}x^{7/3} \).
• Once the antiderivative is found, we use it to evaluate the definite integral by applying the Fundamental Theorem of Calculus.

By determining antiderivatives, we efficiently calculate the accumulated quantity, such as the volume in the Disk Method for our given task.

Solid of Revolution

A Solid of Revolution involves creating a three-dimensional object by revolving a two-dimensional area around a line (often an axis). This concept is essential for understanding how everyday objects like vases or glasses might be designed through mathematical principles.

• To create a solid of revolution, you take a region in a plane and rotate it around an axis – typically the x or y-axis.
• The resulting solid’s volume can be calculated using integral calculus methods like the Disk or Washer Method.
• Understanding the nature of these solids helps us in visualizing how volumes are constructed geometrically and mathematically.

In this problem, the curve \( y = x^{2/3} \) is revolved about the x-axis, forming a solid whose cross-sectional disks are integrated to give the overall volume. This illustrates how a simple function can define a complex three-dimensional shape.