Science:Math Exam Resources/Courses/MATH103/April 2015/Question 03 (a)/Solution 1

The equation for a circle centered at the origin with radius $r$, $x^{2}+y^{2}=r^{2}$, can be written as $y=\pm {\sqrt {r^{2}-x^{2}}}$.

The positive part $y={\sqrt {r^{2}-x^{2}}}>0$ represents the graph for the upper half circle, while the negative part $y=-{\sqrt {r^{2}-x^{2}}}$ corresponds to the graph for the lower half circle.

On the other hand, the geometric meaning of an integral $\int _{a}^{b}f(x)dx$ for the integrand satisfying $f(x)>0$ on $(a,b)$ is the area of the region enclosed by the curves $y=f(x)$, $y=0$, $x=a$, and $x=b$.

Therefore, the area given by $\int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}dx$ corresponds to the upper half disk, so that the answer is as you see in the picture below.

graph of upper-half circle.