Problem 35 - Technical Mathematics with Calculus

Question

Find the moment of inertia of the area bounded by the curve \(y=e^{x},\) the line \(x=1,\) and the coordinate axes, with respect to the \(x\) axis.

Step-by-Step Solution

Answer

\(I_x = \frac{1}{3}(e^{3} - 1).\)

Step 1: Understanding the Area to Consider

The area to consider is bounded by the curve described by the exponential function \(y=e^{x}\), the vertical line \(x = 1\), the x-axis, and the y-axis. This creates a region under the curve from \(x = 0\) to \(x = 1\).

Step 2: Setting Up the Integral for the Moment of Inertia

The moment of inertia \(I_x\) about the x-axis for an area can be calculated using the following integral formula: \[ I_x = \int_\text{area} y^2 \, dA. \] Since \(dA\) is a small area element, for vertical strips parallel to the y-axis, \(dA\) can be written as \(y\,dx\). This leads to the integral: \[ I_x = \int_0^{1} y^2 \, (y\,dx). \] We can substitute \(y = e^{x}\) into this expression to obtain: \[ I_x = \int_0^{1} (e^{x})^2 \, (e^{x}\,dx). \]

Step 3: Calculating the Integral

By simplifying the integral, \[I_x = \int_0^{1} e^{2x} \cdot e^x \, dx = \int_0^{1} e^{3x} \, dx,\] we can find the anti-derivative of \(e^{3x}\), which is \frac{1}{3}e^{3x}. The definite integral will be: \[I_x = \left[ \frac{1}{3}e^{3x} \right]_0^{1} = \frac{1}{3}e^{3} - \frac{1}{3}e^{0}= \frac{1}{3}e^{3} - \frac{1}{3}.\]

Step 4: Simplifying the Result

After performing the subtraction in the expression \( \frac{1}{3}e^{3} - \frac{1}{3} \), we get the final result for the moment of inertia \(I_x\) about the x-axis: \[I_x = \frac{1}{3}(e^{3} - 1).\]

Key Concepts

Understanding Exponential Function Integration

To grasp the concept of integrating an exponential function, we begin by acknowledging that the exponential function, often represented as e^x, has a unique property: it is its own derivative. This means that when we integrate an exponential function, the antiderivative is essentially the same function, often with a multiplicative constant.

For instance, the general form of the antiderivative of e^{kx}, where k is a constant, is (1/k)e^{kx}, plus the constant of integration in an indefinite integral. In the context of the given problem, the integral of e^{3x} with respect to x is (1/3)e^{3x}, since k here is 3. Remember that this integration process is crucial when calculating quantities, like the area under a curve or, as in our case, the moment of inertia.

Understanding how to integrate exponential functions allows students to solve a wide range of problems in calculus, physics, and engineering, making it a fundamental tool in the arsenal of mathematicians and scientists.

Definite Integral Calculation

The calculation of a definite integral involves finding the exact value of an integral over a specific interval. This value represents the net area under the curve described by the function and above the x-axis, taking into account the direction of the area. While an indefinite integral provides the general form of the antiderivative, a definite integral calculates the difference between the values of the antiderivative evaluated at the upper and lower bounds of the interval.

For instance, to determine the definite integral of e^{3x} from x = 0 to x = 1, we evaluate the antiderivative (1/3)e^{3x} at each bound and subtract the lower bound value from the upper bound value. This process is denoted by the Fundamental Theorem of Calculus and is fundamentally tied to the notion of accumulation, which in physics could represent quantities like work, charge, and as relevant to our example, the moment of inertia.

Area Moment of Inertia

The area moment of inertia, commonly denoted as I, is a geometrical property that measures an object's resistance to angular acceleration about a particular axis, in essence reflecting how its mass is distributed with respect to that axis. It is crucial in the design and analysis of structures and mechanical components as it reveals how resistant a section is to bending or twisting.

For areas bounded by functions and axes, the moment of inertia can be calculated using a definite integral where a small element of the area (dA) is multiplied by the square of the distance from the axis we are considering. Specifically, for an area bounded by a curve, the moment of inertia about the x-axis takes the form of I_x = \[ \int y^2 dA \], which represents the integral of the function describing the area (y) squared, over the domain of interest.

Students must appreciate this concept not only to solve textbook exercises but also to understand the physical implications in real-world applications, such as the design of beams, flywheels, and other structural or rotating bodies.