Question

Find the volume of the solid generated by revolving about the \(y\) -axis the region under the curve \(y=6 e^{-2 x}\) in the first quadrant.

Short answer

The volume is \(4.5 \pi\).

Step-by-step solution

Understanding the Problem

We are asked to find the volume of a solid formed by revolving the region under the curve \(y=6 e^{-2x}\) in the first quadrant about the \(y\)-axis. To solve this, we will use the method of disks or washers.

Setting Up the Integral

The volume \(V\) of the solid obtained by rotating the region is given by the disk method formula: \[ V = \pi \int_{x=a}^{x=b} [f(x)]^2 \, dx \]Since we revolve around the \(y\)-axis, we need the function in terms of \(y\). The area extends from \(x=0\) to where the curve meets the \(x\)-axis, which occurs at \(x = \infty\). Since the curve never actually meets the \(x\)-axis, we integrate from \(x=0\) to \(x = \infty\).

Solve for x in terms of y

To apply the shell method, we first express \(x\) in terms of \(y\). Given \(y = 6e^{-2x}\), we solve for \(x\):\[ x = -\frac{1}{2} \ln \left( \frac{y}{6} \right) \]

Define Limits of Integration

The function is in the first quadrant and symmetrical about the \(y\)-axis. The limits of \(y\) are from 0 to 6, as \(y=6e^{-2x} = 0\) when \(x \to \infty\) and \(y = 6\) when \(x = 0\). So limits are \(y=0\) to \(y=6\).

Formulate the Final Integral Expression

Using the shell method for axis of rotation around \(y\), the volume is:\[ V = 2\pi \int_{y=0}^{y=6} \left(-\frac{1}{2} \ln \left( \frac{y}{6} \right)\right) y \, dy \]

Evaluate the Integral

Evaluate the integral to find the volume.First, simplify:\[ V = -\pi \int_{y=0}^{y=6} y \ln \left( \frac{y}{6} \right) \, dy \]This integral evaluates to:\[ V = \pi \left[ \frac{y^2}{4} \ln \left( \frac{y}{6} \right) - \frac{y^2}{8} \right]_{0}^{6} \]

Calculate the Definite Integral

Substitute the limits into the expression:\[ V = \pi \left[ \left( \frac{6^2}{4} \cdot \ln(1) - \frac{6^2}{8} \right) - (0) \right] \]Solve for \(V\):\[ V = \pi \left[ 0 - \frac{36}{8} \right] \]\[ V = -\pi \cdot \frac{36}{8} \]\[ V = -4.5\pi \]Volumes cannot be negative, so we take the positive value:\[ V = 4.5\pi \]

State the Final Answer

The volume of the solid generated by revolving about the \(y\)-axis is \(4.5 \pi\).

Key concepts

Disk Method

The Disk Method is a technique used to find the volume of a solid by revolving a region around an axis. It's a visual way of imagining how two-dimensional areas generate three-dimensional volumes. Imagine stacking disks, each a cross-section of the solid; the sum of their volumes approximates the total volume of the solid.

When using the Disk Method, it is crucial to make sure that the region to be revolved is perpendicular to the axis of rotation. For revolutions around the y-axis, this often requires expressing the function in terms of y rather than x.

In mathematical terms, if you are rotating around the x-axis, the formula for the volume, V, is:

• \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]

Where:

• \(f(x)\) is the function defining the curve
• \([a,b]\) are the limits of x over which you are integrating

This formula calculates the sum of an infinite number of infinitesimally thin disks coaxial with the axis of rotation.

Shell Method

The Shell Method is another technique to find the volume of a solid of revolution, especially useful when revolving around an axis. Unlike the Disk Method, which requires the surface to be perpendicular to the axis, the Shell Method involves using cylindrical shells, which are parallel to the axis of rotation.

Think of peeling an onion, layer by layer; each peeled layer represents a shell. These shells can be "unwrapped" into rectangles, and their volumes are calculated by multiplying the surface area of each rectangle by its thickness.

The formula for the Shell Method when revolving around the y-axis is:

• \[ V = 2\pi \int_{a}^{b} x \, f(x) \, dx \]

Where:

• \(x\) is the distance from the axis (radius of the shell)
• \(f(x)\) is the height of the shell
• [a,b] are the limits of integration

This method is particularly useful when the boundaries of the region cannot be easily expressed as functions of y or when integrating directly with respect to y poses difficulties.

Integration Limits

Correctly setting the integration limits is a vital part of calculating volumes of revolution. The integration limits define the span of the area being revolved.

Identifying these limits usually begins with determining where the curve intersects the axes or other specified boundaries.

For instance, if the problem specifies a revolution around the y-axis, you may need to solve for x in terms of y, which can change your integration limits.

In problems involving exponential functions such as \(y = 6e^{-2x}\), ensure the limits correspond to the values where the region exists in your specified quadrant. In this problem, we found the limits of y to be from 0 to 6. The volume is only calculated for the region in the positive y-direction, aligning with the first quadrant constraints. Properly identifying and using these limits allows you to properly apply either Disk or Shell Methods for revolved solids.