Science:Math Exam Resources/Courses/MATH200/December 2013/Question 05 (a)/Solution 1

We have to start by drawing the following curves to see the domain of integration

$(1)\quad y=0$

$(2)\quad y=1$

$(3)\quad x={\sqrt {3}}y$

$(4)\quad x={\sqrt {4-y^{2}}}$

$(1)$ and $(2)$ are just horizontal lines at $y=0$ and $y=1$ respectively (shown below).

$(3)$ represents a line $y={\frac {x}{\sqrt {3}}}$

$(4)$ represents half a circle of radius $2$ defined for $x>0$. In our case, only the part of the half circle in the first quadrant is necessary because of $(1)$ and $(2)$.

As a result, the domain of integration is the area bounded by these four curves.