Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using geometry.
\[
\int_{-3}^{3} \sqrt{3^{2}-x^{2}} d x
\]
Answer
Final Answer: The value of the integral is approximately \(\boxed{14.14}\).
Step 1 ：The integral represents the area under the curve of the function \(f(x) = \sqrt{3^{2}-x^{2}}\) from \(x = -3\) to \(x = 3\). This function describes a semi-circle with radius 3.
Step 2 ：The area of a semi-circle is given by the formula \(\frac{1}{2} \pi r^{2}\), where \(r\) is the radius of the circle. In this case, \(r = 3\).
Step 3 ：Substituting \(r = 3\) into the formula, we get the area as approximately 14.14 square units.
Step 4 ：This is the value of the integral.
Step 5 ：Final Answer: The value of the integral is approximately \(\boxed{14.14}\).
