91Ó°ÊÓ

The definite integral is a powerful tool used in calculus to calculate the area between a curve and the x-axis over a specific interval. In the provided exercise, our function is given as \( f(x) = \sqrt{1-x^2} \) and the interval is \([0, 1]\). The problem involves finding the area under this curve within the specified range.

To solve this, we establish the integral form \[ A = \int_{0}^{1} \sqrt{1-x^2} \, dx \], which represents the area under \( f(x) \) from \( x = 0 \) to \( x = 1 \). Definite integrals help us not only determine areas but also volumes, central points, and many other mathematical properties.

Key points:

• The concept involves finding a sum of infinite infinitesimal areas under the curve \( f(x) \).
• The limits of integration \([a, b]\) define the interval over which the area is calculated. In our exercise, \( a = 0 \) and \( b = 1 \).
• This type of integral results in a numerical value representing the total area.

Trigonometric substitution simplifies the process of evaluating certain integrals, especially those involving square roots like \( \sqrt{1-x^2} \). In our exercise, we use the substitution \( x = \sin \theta \). This transforms the integrand into a trigonometric function, \( \sqrt{1-x^2} = \cos \theta \), and the differential \( dx \) becomes \( \cos \theta \, d\theta \).

This substitution also changes the limits of integration from \( x = 0 \) and \( x = 1 \) to \( \theta = 0 \) and \( \theta = \frac{\pi}{2} \), respectively. Thus, the integral becomes \[ A = \int_{0}^{\frac{\pi}{2}} \cos^2 \theta \, d\theta \].

Trigonometric substitution benefits include:

• Simplifying integrals that involve the square root of trigonometric identities.
• Converting a function to a more integrable form.
• Utilizing trigonometric identities to further simplify the integral, like using \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \).

The unit circle is a fundamental concept in trigonometry, forming the basis for understanding angles and their corresponding trigonometric ratios. With its center at the origin and a radius of one, it encompasses important properties useful for solving integrals like the one in this exercise.

In the case of the function \( f(x) = \sqrt{ 1 - x^2 } \), we interpret the curve as representing a portion of a semicircle, specifically the upper half of the unit circle. The equation of the unit circle \( x^2 + y^2 = 1 \) gets rearranged to find the y-coordinate for positive \( x \)-values, producing \( y = \sqrt{1-x^2} \). This helps in verifying that the graph represents the top half from \( x = 0 \) to \( x = 1 \), which is pertinent for our area calculation.

Key points about the unit circle include:

• It serves as a reference for understanding circular trigonometry functions and identities.
• The full circle symbolizes all possible angles from 0 to \( 2\pi \) radians, but we focus on a quadrant to solve our problem.
• Understanding the unit circle aids in visualizing and simplifying the integration process.