91Ó°ÊÓ

The definite integral is a fundamental concept in calculus used to compute the area under a curve over a specific interval. In this exercise, we aim to find the area between the graph of the function \(f(x) = \frac{x^2}{\sqrt{1-x^2}}\) and the \(x\)-axis over the interval \([-1/2, 1/2]\). This requires evaluating the definite integral \( \int_{-1/2}^{1/2} f(x) \, dx \).
Definite integrals consist of three main components:

• The function: Here, \(f(x) = \frac{x^2}{\sqrt{1-x^2}}\), represents the curve whose area we are calculating.
• The limits of integration: These are \([-1/2, 1/2]\), specifying the starting and ending points on the \(x\)-axis.
• The result: This provides the total area under the curve from the start to the end points.

By evaluating the definite integral, we sum up tiny slices of area under the curve, providing an exact measure of space the curve covers between these bounds.

Trigonometric substitution is a technique used to simplify the integration of functions involving square roots and quadratic expressions. It leverages trigonometric identities to transform a complex expression into a more manageable form.
In our exercise, to integrate \( \int_{-1/2}^{1/2} \frac{x^2}{\sqrt{1-x^2}} \, dx \), we use trigonometric substitution by setting \(x = \sin(\theta)\). This yields \(dx = \cos(\theta) \ d\theta\) and \(\sqrt{1-x^2} = \cos(\theta)\).
The benefits of trigonometric substitution include:

• Simplification: Replaces complex expressions with easier trigonometric forms.
• Tractable limits: Changes the problem to involve \(\theta\) rather than \(x\).
• Integration ease: Many known trigonometric integrals simplify calculations.

By substituting and changing the limits of integration from \([-1/2, 1/2]\) to \([-\pi/6, \pi/6]\), the integration is considerably simplified to a form involving \(\sin^2(\theta)\).

The concept of the area between curves is essential when you want to find the region bounded by a function and the \(x\)-axis (or between two functions). In this exercise, the goal is to calculate the area under the curve \(f(x) = \frac{x^2}{\sqrt{1-x^2}}\) from \(-1/2\) to \(1/2\).
Here's the approach to find this area:

• Determine Bounds: The function and limits provide bounds of integration.
• Calculate Symmetrically: Since the function \(f(x)\) is even, the area from \([-1/2, 1/2]\) can be halved and evaluated over \([0, 1/2]\), then doubled due to symmetry \(2 \times \int_{0}^{1/2} f(x) \, dx\).
• Compute the Integral: Use techniques like trigonometric substitution to facilitate integration.

Understanding the symmetry and simplifying using trigonometry helped arrive at the final area calculation, ensuring the entire region under the curve is covered accurately. This results in a precise measure of space which is \(\frac{\pi}{3} - \frac{\sqrt{3}}{2}\).