91Ó°ÊÓ

When dealing with definite integrals, one common way to interpret them is by considering the geometric area under a curve. This concept is crucial to understanding how integrals link to actual shapes we can see and work with on a graph. In the original problem, the integral \( \int_{-3}^{0} (1 + \sqrt{9-x^2}) \ dx \) represents the sum of two areas: a rectangle and part of a semicircle. The rectangle derives from the constant function \( 1 \), while the semicircle comes from the expression \( \sqrt{9 - x^2} \). Looking at these areas visually on a graph can greatly aid in comprehension.

When you think of this setup visually:

• The rectangle under the line \( y = 1 \) extends from \( x = -3 \) to \( x = 0 \), creating a straightforward rectangular area.


• The semicircle is a bit more complex, covering the curved part from \( x = -3 \) to \( x = 0 \).

Understanding the area under a curve through components like rectangles and semicircles helps in separating the problem into manageable parts, making it easier to evaluate the definite integral.

The semi-circle equation \( \sqrt{9 - x^2} \) reveals important information about the shape we are dealing with in the provided integral. It stems from the classic circle equation \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle. Here, since \( r = 3 \), we have the circle equation \( x^2 + y^2 = 9 \). By recognizing that \( y = \sqrt{9 - x^2} \) covers only the top part of this circle (where \( y \geq 0 \)), we identify that it forms a semicircle centered at the origin, with the radius reaching 3 units outwards in all directions. This is crucial for solving the integral, as it determines the shape and size of the area we're calculating. Let's break it down:

• The equation \( y = \sqrt{9 - x^2} \) gives the upper semicircle.


• Since the equation is symmetric concerning the y-axis, we're dealing with the left half of the semicircle from \( x = -3 \) to \( x = 0 \).
• The semicircle's radius (3) also gives us crucial details on calculating the area, which employs the formula \( \frac{1}{2}\pi r^2 \) specifically for semicircles.

Understanding the semicircle's equation is thus key in assessing how it interacts with other functions and figuring the area from the definite integral.

In definite integrals, the evaluation over a specific interval is what distinguishes them from indefinite integrals. The interval indicates the region of the x-axis that we are focusing on. For our problem, the interval is defined as \(-3\) to \(0\).

• This interval is particularly significant because it fully covers the left leg of the semicircle given by \( \sqrt{9 - x^2} \), and simultaneously spans the horizontal line \( y = 1 \).


• Evaluating over this interval means finding the exact areas corresponding to these parts and summing them to get the integral's total value.


• Understanding this interval enables us to segment the problem into parts that are easier to analyze, as each segment corresponds to an intuitive geometric shape or figure.

The use of a finite interval does more than just cap the limits of integration. It concretely defines where we measure the area beneath the curve, allowing for precise computation of these geometric areas. Examining the interval helps visualize not only the stretch of the graph being measured but also how different functions contribute to the whole area.