Calculation of Moment of Inertia for the Area Bounded by a Given Curve

In this article, we will delve into the process of calculating the moment of inertia for an area bounded by a specific curve. We will elaborate step-by-step, demonstrating the process with a detailed example and the relevant formulas and integrals used in the calculation. This content is tailored to serve the needs of students, engineers, and professionals dealing with area moments of inertia in their work.

Understanding the Problem: Area Bounded by a Curve

The given curve and boundaries are defined as follows:

xy 3

x 0

y 0

x 3

These boundaries form a triangular region in the xy-plane. We will calculate the moment of inertia of this area with respect to a specific axis, namely the vertical line x 3.

Step 1: Identifying the Area Bounded by the Curve

The first step involves finding the points of intersection to describe the bounded area.

xy 3 intersects the y-axis at:

When x 0, y 3, giving the point (0, 3)

xy 3 intersects the x-axis at:

When y 0, x 3, giving the point (3, 0)

The area bounded by these lines is a triangle with vertices at (0, 0), (0, 3), and (3, 0).

Step 2: Description of the Area

The area can be described as a triangle in the xy-plane with the following dimensions:

Base of the triangle: 3 units (from x 0 to x 3)

Height of the triangle: 3 units (from y 0 to y 3)

Step 3: Calculating the Moment of Inertia

The moment of inertia about the vertical line x 3 is calculated using the formula:

I_x int_A d^2 dA, where d is the distance from the area element dA to the axis of rotation.

For the vertical line x 3, the distance d is given by:

d 3 - x

Step 4: Setting Up the Integral

The area A can be described as follows:

The line y 3 - x gives the height of the triangle at any point x from 0 to 3.

Thus, the area element dA can be expressed as:

dA y dx (3 - x) dx

Step 5: Calculating the Integral for the Moment of Inertia

Now, we can set up the integral for the moment of inertia:

I_x int_0^3 (3 - x)^2 (3 - x) dx

This simplifies to:

I_x int_0^3 (3 - x)^3 dx

Step 6: Evaluating the Integral

Let u 3 - x, then du -dx, and when x 0, u 3, and when x 3, u 0.

So, the integral becomes:

I_x -int_3^0 u^3 du int_0^3 u^3 du

Evaluating the integral:

int u^3 du frac{u^4}{4}

Thus:

I_x left[ frac{u^4}{4} right]_0^3 frac{3^4}{4} - frac{0^4}{4} frac{81}{4}

This means:

The moment of inertia of the area bounded by the given curves with respect to the axis x 3 is:

I_x frac{81}{4}

Summary and Further Exploration

In conclusion, by understanding the area and carefully setting up the integral, we can calculate the moment of inertia of an area bounded by a specific curve. This process can be applied to a wide range of problems in mechanics and structural engineering.

Understanding the area moment of inertia is crucial for many applications in physics and engineering, making it a fundamental concept to master.