Problem 31 - Calculus: Early Transcendentals

Question

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. \( y = e^{-x^2} \) , \( y = 0 \) , \( x = -1 \) , \( x = 1 \) (a) About the x-axis (b) About \( y = -1 \)

Step-by-Step Solution

Answer

(a) For the x-axis: Approximate \( V \) by calculating \( \pi \int_{-1}^{1} e^{-2x^2} \, dx \). (b) For \( y = -1 \): Approximate \( V \) via \( \pi \int_{-1}^{1} [e^{-2x^2} + 2e^{-x^2}] \, dx \).

Step 1: Identify the region

The region of interest is bounded by the curve \( y = e^{-x^2} \), the x-axis \( y = 0 \), and the vertical lines \( x = -1 \) and \( x = 1 \). This region lies between these two curves and within these vertical lines.

Step 2: Set up the integral for (a)

We need to use the disk method to find the volume when rotating about the x-axis. The formula for the volume \( V \) of the solid obtained by rotating around the x-axis is:\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]Here, \( f(x) = e^{-x^2} \), and the bounds are from \( x = -1 \) to \( x = 1 \). So, the integral is:\[ V = \pi \int_{-1}^{1} (e^{-x^2})^2 \, dx = \pi \int_{-1}^{1} e^{-2x^2} \, dx \]

Step 3: Evaluate the integral for (a)

Using a calculator, evaluate the integral \( \pi \int_{-1}^{1} e^{-2x^2} \, dx \). Make sure to choose numerical integration methods if necessary, as this may not have a simple antiderivative.After solving, approximate the result to five decimal places.

Step 4: Set up the integral for (b)

For rotation about the line \( y = -1 \), use the cylindrical shell method or adjust the disk method formula. Use:\[ V = \pi \int_{a}^{b} [(R(x))^2 - (r(x))^2] \, dx \]Here, \( R(x) = e^{-x^2} + 1 \) as the outer radius from \( y = -1 \) to the curve, and \( r(x) = 1 \) as the inner radius to the x-axis. Thus, the volume is:\[ V = \pi \int_{-1}^{1} [(e^{-x^2} + 1)^2 - 1^2] \, dx \]

Step 5: Evaluate the integral for (b)

Using a calculator, evaluate \( \pi \int_{-1}^{1} [(e^{-x^2} + 1)^2 - 1^2] \, dx \). This simplifies to:\[ V = \pi \int_{-1}^{1} [e^{-2x^2} + 2e^{-x^2}] \, dx \]Again, calculate the integral numerically to obtain an approximate result correct to five decimal places.

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of calculus that deals with the concept of integration. It's a method used to find quantities like area, volume, and other sums through the accumulation of small, infinitely-sized parts. Integration undoes differentiation and is used to calculate total values from rates of change.

When solving problems involving solids of revolution, we apply integral calculus to compute the volume of a solid shape generated by rotating a region around a line (an axis). To set up an integral for volume, it's crucial to identify:

• The function, or curve, describing the region
• The bounds of integration
• The axis of rotation

This informs us how the region's parts add up to form the complete solid. The integration technique can vary based on these factors and often leads to numerical solutions when exact antiderivatives are unavailable.

Disk Method

The disk method is a technique used in integral calculus to find the volume of a solid of revolution. It involves slicing the solid perpendicular to the axis of rotation into thin cylindrical disks. These disks resemble coins or pancakes, which have both a small height and a circular surface.

To apply the disk method, we typically use the formula:
\[ V = \,\pi \int_{a}^{b} [f(x)]^2 \, dx \]
Here's how it works:

• The function \( f(x) \) represents the radius of each disk.
• Each disk contributes a small volume increment, \( \pi [f(x)]^2 \, dx \), to the total volume.
• Integrating over the interval \([a, b]\) sums all these increments.

In the problem, when the region under \( y = e^{-x^2} \) is rotated around the x-axis, \( f(x) = e^{-x^2} \) defines the radius of the disks. The volume involves computing \( \pi \int_{-1}^{1} e^{-2x^2} \, dx \). This integral doesn't have a simple form, often requiring numerical methods for evaluation.

Cylindrical Shell Method

The cylindrical shell method is another approach in integral calculus to find the volume of solids of revolution. It's particularly useful when the axis of rotation isn't one of the main axes. It considers the solid composed of concentric cylindrical shells.

The general formula for the volume using the shell method is:
\[ V = 2 \pi \int_{a}^{b} x \cdot f(x) \, dx \] (when rotating around the y-axis)
In our exercise, we instead adjusted the disk method to accommodate rotation around the line \( y = -1 \), demonstrating a flexible approach for different scenarios. Here, the radius is redefined where:

• \( R(x) = e^{-x^2} + 1 \) (outer radius)
• \( r(x) = 1 \) (inner radius)

The setup becomes \( \pi \int_{-1}^{1} [(e^{-x^2} + 1)^2 - 1^2] \, dx \), simplifying further into \( \pi \int_{-1}^{1} [e^{-2x^2} + 2e^{-x^2}] \, dx \). Each shell calculates part of the solid's volume, which you finally sum using integration.

Numerical Integration

Numerical integration provides approximate solutions when exact antiderivatives are impossible to find or complex to compute. Even with intricate functions or non-standard forms, numerical methods help in accurately estimating integrals.

Some common numerical techniques include:

• Trapezoidal Rule
• Simpson's Rule
• Newton-Cotes Formulas

These methods partition the integration range into small subintervals, approximating the area under the curve piece by piece. It uses weighted sums that approach the integral's value.

In exercises like rotating a function \( y = e^{-x^2} \) about an axis, it's often necessary to apply numerical integration due to the lack of straightforward antiderivatives. Calculators or specialized software can implement these techniques, delivering results accurate to high decimal places—essential for precision in complex calculations.