Area of Region

**Mrdavid445** · August 9th 2012, 04:35 PM

What is the area of the region in the first quadrant bounded by the x-axis, the line $y=2x$ , and the circles $x^2+y^2=20$ and $x^2+y^2=30$ ?

**GJA** · August 9th 2012, 07:01 PM

Hi, Mrdavid445.

The area of a region R in the plane is given by

$\int\int_{R}1\cdot dA$ ,

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

$dA=rdrd\theta$ .

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!

**earboth** · August 9th 2012, 11:01 PM

Originally Posted by Mrdavid445

What is the area of the region in the first quadrant bounded by the x-axis, the line $y=2x$ , and the circles $x^2+y^2=20$ and $x^2+y^2=30$ ?

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:

$r_1 = \sqrt{20} \approx 4.472$

$r_2 = \sqrt{30} \approx 5.477$

$|\theta| = \arctan(2) \approx 63.435^\circ$

Draw a sketch!

2. The area a is calculated by:

$a = \frac{|\theta|}{360^\circ} \cdot \pi r_2^2 - \frac{|\theta|}{360^\circ} \cdot \pi r_1^2 =\frac{|\theta|}{360^\circ} \cdot \pi (r_2^2 - r_1^2)$

Plug in the known values.

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