91Ó°ÊÓ

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is especially useful in integration when the expressions involve square roots or are part of a circle's equation. To complete the square for the expression \( 6x - x^2 \), we first rearrange the terms as \( -(x^2 - 6x) \). Next, find half of the coefficient of \( x \) in \( x^2 - 6x \), which is \( 3 \), and then square it to get \( 9 \). By adding and subtracting this square inside the parentheses, we get:

• \( -(x^2 - 6x + 9) + 9 = (x-3)^2 - 9 \)

Rewrite the original expression as \( 9 - (x-3)^2 \). This form makes it easier to identify patterns or symmetries, such as those found in circle equations. When completed, the square allows the integral to be evaluated using geometric insights and proper formulas.

Circle geometry deals with the properties and equations that describe circles and arcs. In the context of the given integral, the expression \( \sqrt{9 - (x-3)^2} \) corresponds to a semicircle equation centered at the point \((3,0)\) with a radius of \(3\). Here’s why that matters:

• The general form of a circle's equation is \( (x-h)^2 + y^2 = R^2 \), where \((h,0)\) is the center and \(R\) is the radius.
• The expression \(R^2 - (x-h)^2\) defines a semicircle along the y-axis, opening upwards or downwards, with \( y = \sqrt{R^2 - (x-h)^2} \) describing the upper half.

In our case, since the circle is centered at (3,0) and has a radius of 3, the expression represents the upper half of the circle between \(x = 0\) and \(x = 3\). Understanding this geometric interpretation helps us to apply the right formulas for area calculation.

Evaluating integrals through geometry is an efficient method, especially for functions representing circular arcs or other symmetrical shapes. For the integral \( \int_{0}^{3} \sqrt{9 - (x-3)^2} \, dx \), the function under the square root describes the upper half of a circle centered at \((3,0)\) with a radius of 3. Here's how we can evaluate it:

• Calculate the area of the entire circle: \( \pi R^2 = \pi \times 3^2 = 9\pi \).
• Recognize that the integral represents a quarter of this circle's area since it only covers the segment from \(x = 0\) to \(x = 3\), which is a quadrant of the full circle.
• The area of this quarter circle is given by \( \frac{1}{4} \times 9\pi = \frac{9\pi}{4} \).

By utilizing the properties of the circle, we save time and avoid complex calculations, accurately finding the integral's value through geometric principles.