Question

exercises 1.2 the definite integral
score: 40/170 answered: 4/17
question 5
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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_{-6}^{6} \sqrt{6^2 - x^2} dx\\)
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Explanation:

Step1: Recognize the function's graph

The function $y=\sqrt{6^2 - x^2}$ rearranges to $x^2 + y^2 = 6^2$ with $y\geq0$, which is the upper half of a circle with radius $r=6$.

Step2: Identify the integral's interval

The integral spans $x=-6$ to $x=6$, covering the full horizontal range of the semicircle.

Step3: Calculate the semicircle area

Use the area formula for a semicircle: $\frac{1}{2} \pi r^2$. Substitute $r=6$:
$\frac{1}{2} \pi (6)^2 = \frac{1}{2} \pi \times 36$

Answer:

$18\pi$