What is the area of the region in the first quadrant bounded by the x-axis, the line $\displaystyle y=2x$, and the circles $\displaystyle x^2+y^2=20$ and $\displaystyle x^2+y^2=30$?
Hi, Mrdavid445.
The area of a region R in the plane is given by
$\displaystyle \int\int_{R}1\cdot dA$,
where dA is the area element. Since the region we're interested in is a portion of an annular region, it's best to use the polar form for dA, which is
$\displaystyle dA=rdrd\theta$.
What remains are to determine the limits of integration and then compute the double integral.
Does this get everything going in the right direction?
Good luck!
Since you posted your question in the Geometry forum you probably wanted to use a more geometrical way to solve the problem(?).
1. The area in question is the difference of 2 sectors of 2 concentric circles:
$\displaystyle r_1 = \sqrt{20} \approx 4.472$
$\displaystyle r_2 = \sqrt{30} \approx 5.477$
$\displaystyle |\theta| = \arctan(2) \approx 63.435^\circ$
Draw a sketch!
2. The area a is calculated by:
$\displaystyle a = \frac{|\theta|}{360^\circ} \cdot \pi r_2^2 - \frac{|\theta|}{360^\circ} \cdot \pi r_1^2 =\frac{|\theta|}{360^\circ} \cdot \pi (r_2^2 - r_1^2)$
Plug in the known values.