Question

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 high school geometry. (int_{-3}^{3}sqrt{9 - x^{2}}dx)

Explanation:

Step1: Identify the function's graph

The function $y = \sqrt{9 - x^{2}}$ can be rewritten as $y^{2}=9 - x^{2}$, or $x^{2}+y^{2}=9$ with $y\geq0$. It represents the upper - half of a circle centered at the origin with radius $r = 3$.

Step2: Recall the area formula for a semi - circle

The area formula for a full circle is $A=\pi r^{2}$. For a semi - circle (upper or lower half), the area formula is $A=\frac{1}{2}\pi r^{2}$.

Step3: Calculate the area

Given $r = 3$, substituting into the semi - circle area formula $A=\frac{1}{2}\pi r^{2}$, we get $A=\frac{1}{2}\pi(3)^{2}=\frac{9\pi}{2}$.

Answer:

$\frac{9\pi}{2}$