91Ó°ÊÓ

In calculus, a definite integral represents the area under a curve between two specified points. It computes the accumulation of quantities, such as distance or area, across an interval. The notation for a definite integral is \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration, and \(f(x)\) is the function to be integrated.

Key points related to definite integrals include:

• The integral is definite because it is evaluated over a specific interval.
• It gives a numerical value, unlike an indefinite integral which results in a function plus a constant.
• The Fundamental Theorem of Calculus connects the concept of derivatives with integrals.

The exercise involves a definite integral \( \int_{1 / 2}^{1} \frac{6 \, dt}{\sqrt{3+4t-4t^{2}}} \), where the goal is to find the area under the specified curve from \( t = \frac{1}{2} \) to \( t = 1 \). By transforming the integrand into a manageable form, we successfully calculate this area.

Trigonometric substitution is a technique for solving integrals involving square roots, especially those of the form \( a^2 - x^2 \). By substituting a trigonometric function, these integrals become easier to handle because of the Pythagorean identities. For instance, the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) plays a crucial role here.

The exercise demonstrates this method by substituting \( t - \frac{1}{2} = \sin(\theta) \). This substitution simplifies the integrand by eliminating the square root via the identity \( 1 - \sin^2(\theta) = \cos^2(\theta) \). Important aspects include:

• Choosing the correct substitution that matches the pattern \( a^2 - x^2 \).
• Transforming the limits of integration to match the new variable.
• Expressing \( dt \) in terms of \( d\theta \) to complete the substitution.

This technique linearizes otherwise complex integrals and streamlines the process of resolving them.

Completing the square is a method used to transform a quadratic expression into a perfect square. This is particularly useful when the expression is under a square root in an integrand, as simplifying the expression paves the way for subsequent techniques like trigonometric substitution.

In our exercise, the expression \( 3 + 4t - 4t^2 \) is converted into a recognizable form through square completion:

• Reorganize the terms to isolate the quadratic and linear components: \( -4(t^2 - t) \).
• Add and subtract \( \left(\frac{1}{2}\right)^2 \) within the expression to achieve the perfect square: \( (t-\frac{1}{2})^2 - 1 \).
• Multiply through by factors outside the square root for simplification: \( 4(1-(t-\frac{1}{2})^2) \).

This transformation makes it feasible to apply trigonometric identities and substitutions effectively in the integration process.

After transforming the integral into a solvable form, the next step is to evaluate it. In our worked example, the integral simplifies from a trigonometric substitution into a much simpler expression.

The key elements of integral evaluation include:

• Substituting back the simplified expression: \( \int_{0}^{\pi/6} 3 \, d\theta \).
• Using basic integral rules to evaluate: The integral of a constant \(3\) with respect to \(\theta\) results in \(3\theta\).
• Applying the limits of integration to find the final value: \( 3\theta \Big|_0^{\pi/6} \).

By carefully managing these steps, the calculated result is \( \frac{\pi}{2} \), representing the area under the curve per the original problem statement.