Question

Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. $$ y=e^{-x} y=0, x=-1 x=0 ; \text { about } x=1 \text { (Express the answer in exact form.) } $$

Short answer

Use cylindrical shells and integrate from -1 to 0: \( V = 2\pi \int_{-1}^{0} (1-x)e^{-x} \, dx \).

Step-by-step solution

Determine the Bounded Region

The region bounded by the curves is between the curve \( y = e^{-x} \), the line \( y = 0 \) (the x-axis), and the vertical lines \( x = -1 \) and \( x = 0 \). This is a strip that extends vertically from the x-axis to the curve between these x-values.

Set Up the Volume Integral

To find the volume when rotated around the line \( x = 1 \), use the method of cylindrical shells. The formula for the volume is \[ V = 2\pi \int_{a}^{b} (radius)(height) \, dx \] where the radius is the distance from the line \( x = 1 \) to \( x \), i.e., \( 1-x \), and the height is \( y = e^{-x} \).

Key concepts

Volume of Revolution

Creating a volume by rotating a region around a line is an essential concept in calculus known as the "Volume of Revolution." Imagine taking a two-dimensional area and spinning it around a specified line. This action forms a three-dimensional shape, and finding its volume involves evaluating how much space this solid occupies. This concept can be visualized with everyday objects like a vase. Drawing its profile on paper and spinning it around its axis yields its full three-dimensional volume.

In the given exercise, we're rotating a region bounded by the curve and specific lines about the line \( x = 1 \). We're essentially calculating how much space the resultant shape -- formed by this rotation -- takes up. Understanding this idea is pivotal for various applications in engineering and physics, such as computing the volume of a tank or a dome.

Cylindrical Shells Method

The Cylindrical Shells Method is a powerful technique in calculus for finding the volume of a solid of revolution. It is particularly useful when the axis of rotation is vertical, as in our problem. The technique uses imaginary cylindrical shells to simplify the process of calculating volume.

• Think of each shell as a thin-walled cylinder.
• The volume of a thin shell is equal to its circumference times its height times its thickness.

To apply this method, we use the integral formula:\[V = 2\pi \int_{a}^{b} (radius)(height) \, dx\]Here, 'radius' is the distance from the axis of rotation, \( x = 1 \), to a point on the x-axis. In this exercise, it's \( 1-x \). The 'height' is the vertical distance described by the function, \( e^{-x} \).

This method shines when the curve and the axis of rotation are not aligned. It also often requires less computation than the "disk" or "washer" methods, especially when the region doesn’t revolve around the x-axis or y-axis directly.

Exponential Function

The curve involved in this exercise is defined by an exponential function: \( y = e^{-x} \). An exponential function is a mathematical function that grows or decays at a rate proportional to its current value. This specific function is a decaying exponential because of the negative exponent.

• As \( x \) increases, \( e^{-x} \) decreases towards zero.
• Defined for all \( x \), it never actually hits zero, illustrating an asymptotic behavior.

In the realm of volume calculation, exponential functions often provide elegant solutions because their rate of change lends well to integrating processes.

Exponential decay, like \( e^{-x} \), often appears in real-world contexts such as in physics for radioactive decay, or even in the spread of diseases where each successive unit of time might see a proportional decrease in new cases.

Integral Calculus

Integral calculus is the branch of calculus focused on accumulation. It helps us find quantities like areas, volumes, and other significant measurements. In this problem, we're using integral calculus to determine the volume of a solid of revolution. This process is grounded in summing infinite tiny slices that approximate the whole volume.At the core of integral calculus is the integral, which, in simple terms, is a mathematical tool for adding up these small parts. Specifically, we express the volume as an integral:\[V = 2\pi \int_{-1}^{0} (1-x)e^{-x} \, dx\]This integral calculates the sum of an infinite number of cylindrical shells from \( x = -1 \) to \( x = 0 \). By understanding how to apply integral calculus, you gain the ability to tackle not just this problem but many others where accumulation is key. Whether you're computing the area under a curve, the length of a curve, or the volume of complex shapes, integral calculus provides the mathematical framework to solve these intricate problems.