Problem 61 - Calculus Single Variable

Question

In each of Exercises 61-64, use the method of disks to calculate the volume obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region between the curves \(y=\exp (x)\), the \(y\) -axis, and the line \(y=e\).

Step-by-Step Solution

Answer

The volume of the solid is \(\frac{\pi}{2} (e^2 - 1)\).

Step 1: Understand the Region 

The region al{R} is bounded by the curve \(y = \exp(x)\), the y-axis \((x = 0)\), and the horizontal line \(y = e\). \(y = \exp(x)\) represents an exponential growth starting at \(y = 1\) when \(x = 0\). Thus, it intersects the line \(y = e\) at \(x = 1\). Hence, the region is bounded between \(x = 0\) and \(x = 1\).

Step 2: Set Up the Disk Method Formula 

The formula for volume using the method of disks when rotating around the \(y\)-axis is: \[ V = \int_{a}^{b} \pi \left(f(x)\right)^2 \, dx \] Here, \(f(x) = \exp(x)\) and the limits of integration are from \(x = 0\) to \(x = 1\).

Step 3: Substitute and Simplify 

Substitute \(f(x) = \exp(x)\) into the formula: \[ V = \int_{0}^{1} \pi (\exp(x))^2 \, dx = \pi \int_{0}^{1} \exp(2x) \, dx \]Since \((\exp(x))^2 = \exp(2x)\).

Step 4: Integrate the Function 

Find the integral of \(\exp(2x)\): \[ \int \exp(2x) \, dx = \frac{1}{2} \exp(2x) + C \]Using the limits of integration, evaluate the definite integral: \[ V = \pi \left[ \frac{1}{2} \exp(2x) \right]_{0}^{1} \]

Step 5: Evaluate the Definite Integral 

Substitute the limits into the antiderivative:\[ V = \pi \left( \frac{1}{2} \exp(2 \times 1) - \frac{1}{2} \exp(2 \times 0) \right) \]\[ V = \pi \left( \frac{1}{2} \exp(2) - \frac{1}{2} \right) \]

Step 6: Calculate the Volume 

Simplify the expression:\[ V = \pi \left( \frac{1}{2} e^2 - \frac{1}{2} \right) = \pi \left( \frac{1}{2} (e^2 - 1) \right) \]\[ V = \frac{\pi}{2} (e^2 - 1) \]

Key Concepts

Method of Disks

The method of disks is a powerful technique used to calculate the volume of a solid of revolution. When a region in the plane is rotated around an axis, this method helps in determining the volume of the resulting three-dimensional object. It breaks down the volume into an infinite number of thin, flat disks stacked along the axis of rotation. Each disk has a small thickness and a radius dependent on the function describing the boundary of the region.

To use the method of disks:

• First, determine the bounds of the region to be rotated.
• Identify the axis of rotation. In our exercise, this is the y-axis.
• The volume is calculated using the integral formula: \[ V = \int_{a}^{b} \pi (f(x))^2 \; dx \] where \( f(x) \) represents the radius of each disk, which in this rotational context, is the distance from the y-axis to the curve.

This approach essentially sums up the volumes of all the disks from the starting to the ending point of the domain along the axis of rotation.

Exponential Functions

Exponential functions are a class of mathematical functions characterized by a constant base raised to a variable exponent. They are generally written in the form \( f(x) = a \cdot b^x \) where \( a \) and \( b \) are constants. In the context of this exercise, the exponential function is \( y = \exp(x) \), which is a common way to denote \( y = e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Exponential functions exhibit rapid growth (or decay, if the exponent is negative) and are fundamental in modeling a variety of natural and theoretical phenomena:

• As x increases, \( \exp(x) \) grows swiftly, demonstrating exponential growth.
• When rotated around the y-axis, an exponential curve creates an interesting volume due to its continuously increasing radius.

In the specific case of rotating \( y = \exp(x) \) about the y-axis, the radius of each disk formed during rotation is simply this exponential function itself.

Definite Integrals

Definite integrals are a core concept in calculus used to calculate the accumulation of quantities, such as areas under curves, average values, and volumes of solids, across a given interval. When we apply definite integrals in the context of the method of disks, we are summing up an infinite number of infinitely thin slices to find the total volume.

The process involves:

• Finding an antiderivative of the function involved.
• Evaluating this antiderivative at the boundaries of the interval.
• Subtracting the resulting values, which gives the accumulated quantity over the interval.

In the provided solution, the definite integral of \( \exp(2x) \) is calculated from \( x = 0 \) to \( x = 1 \), yielding the definite volume of the solid:\[V = \pi \left( \frac{1}{2} e^2 - \frac{1}{2} \right)\] This is the final step to determine not only how the function accumulates but also its geometrical significance in constructing volumes.