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The area of a plane region \( R \) as a double integral is

  • A.\( \displaystyle\iint_{R} (x + y)\,dA \)
  • B.\( \displaystyle\iint_{R} xy\,dA \)
  • C.\( \tfrac{1}{2}\displaystyle\iint_{R} dA \)
  • D.\( \displaystyle\iint_{R} dA \)

Answer: D

Taking the integrand as one makes each element contribute its own area, so the total is the area itself:

\[ A \;=\; \iint_{R} dA. \]

An integrand of \( x \) or \( y \) gives a first moment instead, from which a centroid is found, and a product \( xy \) gives a product moment

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