Question

exercises 1.3 the fundamenta score: 140/260 answered: 14/26 question 15 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} dx\\) question help: video message instructor submit question jump to answer

Explanation:

Step1: Identify the function's graph

The function $y=\sqrt{3^2 - x^2}$ rearranges to $x^2 + y^2 = 9$ with $y\geq0$, which is the upper half of a circle with radius $r=3$.

Step2: Match integral to geometric area

The integral $\int_{-3}^{3} \sqrt{3^2 - x^2}dx$ represents the area of the upper semicircle (since $x$ ranges from $-3$ to $3$, covering the full horizontal span of the circle).

Step3: Calculate the semicircle area

The area of a full circle is $\pi r^2$, so the area of a semicircle is $\frac{1}{2}\pi r^2$. Substitute $r=3$:
$\frac{1}{2}\pi (3)^2 = \frac{9}{2}\pi$

Answer:

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