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 geometry.\\(\int_{-2}^{2} \sqrt{2^2 - x^2} dx\\)

Explanation:

Step1: Recognize the function's graph

The function $y = \sqrt{2^2 - x^2}$ rearranges to $x^2 + y^2 = 4$ with $y \geq 0$, which is the upper half of a circle with radius $r=2$.

Step2: Identify the integral's bounds

The integral spans $x=-2$ to $x=2$, covering the full diameter of the semicircle.

Step3: Calculate the semicircle area

Use the circle area formula, take half:
$$\text{Area} = \frac{1}{2} \pi r^2$$
Substitute $r=2$:
$$\text{Area} = \frac{1}{2} \pi (2)^2 = 2\pi$$

Answer:

$2\pi$