Question

In Exercises 21-30, sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral \((a>0, r>0)\) $$ \int_{-3}^{3} \sqrt{9-x^{2}} d x $$

Short answer

The value of the integral is \(4.5\pi\).

Step-by-step solution

Identify the Shape

The function \(\sqrt{9-x^{2}}\) is a semicircle with a radius of 3 above the x-axis. If we plot the function, we'll get the upper half of a circle with a radius of 3.

Formulate the Area

Since we know the shape is a semi-circle, we can use the geometric formula for the area of a semi-circle to calculate the result. The formula for the area of a semi-circle is \(\frac{1}{2} \pi r^{2}\), where \(r\) is the radius. Thus, the area of our semi-circle is \(\frac{1}{2} \pi 3^{2}\)

Calculate the Area

Insert the value of \(r=3\) to get the area: \(\frac{1}{2} \pi (3)^{2} = \frac{1}{2} \pi 9 = 4.5\pi\). This is the value of our definite integral.

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on the accumulation of quantities and the areas beneath curves. At its heart lies the fundamental concept of the integral, which not only measures the area under a curve on a graph but also represents a wide variety of physical quantities such as distances, volumes, and mass.

In the context of a definite integral, which has specific upper and lower limits, the goal is to determine the total accumulation between those points. This process is analogous to adding up an infinite number of infinitesimally thin rectangles beneath the curve to find the exact area. The formal definition involves a limit process that leads to the integral we're familiar with: \[ \int_{a}^{b} f(x) dx \.\]
This expression tells us to calculate the area under the function \( f(x) \) from \( x = a \) to \( x = b \). In the exercise provided, the definite integral represents the area under the semicircle defined by \( \sqrt{9-x^{2}} \) from \( x = -3 \) to \( x = 3 \).

Area Under a Curve

The area under a curve in a particular interval can be visualized as the space between the curve itself and the x-axis within the range of integration. Calculating this area allows us to understand the integral better and connect it to real-world quantities.

For instance, the equation \( f(x)=\sqrt{9-x^{2}} \) represents the top half of a circle, which is essentially a semicircle. The integral given in the exercise, \[ \int_{-3}^{3} \sqrt{9-x^{2}} dx \.\] seeks the area of this semicircle from \( x = -3 \) to \( x = 3 \).

By sketching the function on a graph, a visual representation of the area can aid in understanding the abstract concept of integration. The curve forms an arc, and the x-axis serves as the base, enclosing the area in question. The height at any point in the interval equates to the y-value of the function, \( f(x) \).

Geometric Formula

A geometric formula describes a rule or statement that relates to the shape, size, or properties of a figure or object in a mathematical context. These formulas are particularly helpful when we deal with definite integrals that correspond to familiar geometric shapes.

The geometric formula for the area of a circle is \( \pi r^{2} \), so for a semicircle (which is half of a circle), it becomes \( \frac{1}{2} \pi r^{2} \). Knowing this, evaluating the area of the semicircle becomes straightforward if we are given the radius. For the given exercise, the radius of the semicircle is 3, hence the area is \{ \frac{1}{2} \pi (3)^{2} = \frac{1}{2} \pi 9 = 4.5\pi \}.

Understanding and applying these formulas are key to solving integrals geometrically. This approach provides an alternative to the algebraic method of evaluating integrals, which is especially useful in cases where the integral corresponds to a well-known shape.