Problem 6 - Calculus: Early Transcendentals

Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( 2x = y^2 \) , \( x = 0 \) , \( y = 4 \) ; about the y-axis

Step-by-Step Solution

Answer

The volume is \( \frac{256\pi}{5} \).

Step 1: Set Up the Problem

We are given the curves \( 2x = y^2 \), \( x = 0 \), and \( y = 4 \). We need to find the volume of the solid formed by rotating the region enclosed by these curves about the y-axis. We shall use the disk method here because the solid is formed by rotating around the y-axis.

Step 2: Express x in terms of y

The given equation \( 2x = y^2 \) can be solved for \( x \):\[ x = \frac{y^2}{2} \].

Step 3: Determine the bounds of integration

The curves intersect at \( x = 0 \) and \( 2x = y^2 \). Since we are considering the region bounded by these curves, and the line \( y = 4 \), the limits of \( y \) go from \( 0 \) to \( 4 \).

Step 4: Set up the volume integral

Using the disk method, for rotation about the y-axis, the volume \( V \) is given by:\[ V = \pi \int_{c}^{d} [R(y)]^2 \, dy \]where \( R(y) \) is the radius of a typical disk. In this problem, \( R(y) = \frac{y^2}{2} \). So, the volume integral is:\[ V = \pi \int_{0}^{4} \left(\frac{y^2}{2}\right)^2 \, dy \].

Step 5: Evaluate the integral

Simplify the integrand:\[ \left(\frac{y^2}{2}\right)^2 = \frac{y^4}{4} \].Plug into the integral:\[ V = \pi \int_{0}^{4} \frac{y^4}{4} \, dy \].Factor out the constant \( \frac{1}{4} \):\[ V = \frac{\pi}{4} \int_{0}^{4} y^4 \, dy \].

Step 6: Calculate the integral

Integrate \( y^4 \): \[ \int y^4 \, dy = \frac{y^5}{5} \].Apply the limits of integration:\[ V = \frac{\pi}{4} \left[ \frac{y^5}{5} \right]_{0}^{4} = \frac{\pi}{4} \left( \frac{4^5}{5} - \frac{0^5}{5} \right) \].

Step 7: Final computation

Calculate \( 4^5 = 1024 \), then substitute:\[ V = \frac{\pi}{4} \times \frac{1024}{5} = \frac{1024 \pi}{20} = \frac{256 \pi}{5} \].

Step 8: Conclusion

The volume of the solid obtained by rotating the given region about the y-axis is \( \frac{256\pi}{5} \).

Key Concepts

Disk Method

The disk method is a powerful technique used in calculus to find the volume of solids of revolution. When a region in the plane is rotated around a line, it forms a 3-dimensional solid. The disk method simplifies this complex shape into a series of thin disks or rings stacked along the axis of rotation. Each of these disks looks similar to a pancake or a hockey puck.
To apply the disk method, you must first determine the radius of each disk. The radius usually depends on the function that forms the boundary of the region being rotated.

• The volume of each disk is related to its area, which is given by \( \pi R^2 \).
• Integrating the area of these disks from one boundary to another gives the total volume.

Using the disk method, the challenge is to properly establish the limits of integration and the expression for the radius, as it often involves converting equations to suitable forms for integration.

Volume of Solids of Revolution

The concept of the volume of solids of revolution revolves around generating three-dimensional volumes by rotating a two-dimensional area about an axis. Consider a flat shape, like a piece of paper, spinning around a pencil; it forms a 3D object.
This method works for several shapes, including those bounded by curves or areas defined between specific boundaries.

• Typically, the methods used include the disk or washer method, and sometimes the shell method, depending on the axis and setup.
• The key is to set up the correct integral that sums the incremental volumes contributed by rotation about the axis.

In this exercise, rotating the given region around the y-axis requires careful consideration of these aspects to properly calculate the resulting solid's volume.

Definite Integral

Definite integrals are a fundamental concept in calculus, serving not only to find areas under curves but also volumes and other cumulative measures. In this context, the definite integral computes the combined volume of an infinite series of thin disks.

The integral expression, typically \( \int_a^b f(x) \, dx \), involves:

• Setting the limits of integration, here determined by the bounds of the region being rotated (in this case, y = 0 to y = 4).
• Calculating the expression inside the integral which corresponds to the area of the disks at any given cross-section along the axis.

The solved example transforms the function into a more suitable format for integration, ensuring errors from manual or conceptual misunderstandings are minimized.

Y-Axis Rotation

Rotating around the y-axis creates a solid that is symmetric along the vertical axis. This axis of rotation provides unique advantages when using calculus to find volumes.

When calculating rotations around the y-axis:

• The problem often involves expressing one variable in terms of another to accommodate the integration with respect to the vertical y-axis.
• It's crucial to ensure that the radius of disks or shells are consistent with their positioning relative to this axis.

As demonstrated in the solution, expressing x in terms of y simplifies integration, enabling a straightforward application of the chosen method to find the volume. This rotation simplifies integrating perpendicular slices of the solid, providing a clear path to the final volume calculation.