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Math Help - Rectangle

  1. #1
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    Rectangle

    A rectangle is bounded by the x-axis and the semicircle
    y = sqrt[36 - x^2] . Write the area A of the rectangle as a function of x, and determine the domain of the function.

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  2. #2
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    I assume that the circle is centred at the origin.

    Clearly, its domain is -6 \leq x \leq 6 and its range is 0 \leq y \leq 6.

    The coordinate of any point on the semicircle is \left(x, \sqrt{36 - x^2}\right).

    Therefore, as you move the point along the semicircle, its length is x in each direction of the horizontal, and it gains a length of \sqrt{36 - x^2} on its vertical.

    So the length of the rectangle is 2x and the width is \sqrt{36 - x^2}.

    Therefore its area is

    A = 2x\sqrt{36 - x^2} where 0 < x< 6.
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  3. #3
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    Beautifully explained.
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