Find the area of the region that lies beneath the given curve. $$y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$$
A) $1.500$
B) $1.450$
C) $- 0.500$
D) $- 1.500$
E) $0.500$

Question

Quizplus · Accepted Answer

We can set up the integral for the area as $\int_0^{\frac{\pi}{3}}\sin x\,dx$, and evaluate it to get $\boxed{0.500}$. Note that since $\sin x$ is always non-negative on the given interval, the area cannot be negative, so we can eliminate choices C and D. Additionally, the answer cannot be greater than $\frac{\pi}{6}$ (which is the area of the region bounded by the $x$-axis and the line $y=\frac{1}{2}$), so we can eliminate choice A. Finally, the answer cannot be greater than $\frac{\sqrt{3}}{4}$ (which is the area of the region bounded by the $x$-axis and the equilateral triangle with vertices at $(0,0)$, $(\frac{\pi}{3},0)$, and $(\frac{\pi}{6},\frac{\sqrt{3}}{6})$), so we can eliminate choice B. Thus, the only possible answer is E, and evaluating the integral confirms that it is indeed the correct answer.

Find the Area of the Region That Lies Beneath the Given