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Math Help - Area of Region

  1. #1
    Junior Member Mrdavid445's Avatar
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    Area of Region

    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?
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  2. #2
    GJA
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    Re: Area of Region

    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!
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    Re: Area of Region

    Quote Originally Posted by Mrdavid445 View Post
    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.
    Attached Thumbnails Attached Thumbnails Area of Region-sektordifferenz.png  
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