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Reversing the order in \( \displaystyle\int_{0}^{1}\!\!\int_{x}^{1} f\,dy\,dx \) gives limits

  • A.\( \displaystyle\int_{0}^{1}\!\!\int_{0}^{1} f\,dx\,dy \)
  • B.\( \displaystyle\int_{0}^{1}\!\!\int_{0}^{y} f\,dx\,dy \)
  • C.\( \displaystyle\int_{0}^{1}\!\!\int_{y}^{1} f\,dx\,dy \)
  • D.\( \displaystyle\int_{x}^{1}\!\!\int_{0}^{1} f\,dx\,dy \)

Answer: B

The region is the triangle with \( 0 \le x \le 1 \) and \( x \le y \le 1 \), that is the part of the unit square above the line \( y = x \). Describing it the other way, \( y \) runs from zero to one and for each \( y \) the variable \( x \) runs from zero up to \( y \):

\[ \int_{0}^{1}\!\!\int_{0}^{y} f\,dx\,dy. \]

Sketching the region is the only reliable way to get this right

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