Discussion
The area of the region bounded by \(y = x^{5/2}\) and \(y = x\) (in square units) is
Answer: C
The area is \(\frac{3}{14}\) square units.
\(x^{5/2}\) is defined for \(x \geq 0\), and the curves meet where \(x^{5/2} = x\), at \(x = 0\) and \(x = 1\). Between them the line lies above the curve:
\[ \begin{aligned} A &= \int_{0}^{1}\left(x - x^{5/2}\right)dx \\ &= \frac{1}{2} - \frac{2}{7} = \frac{3}{14} \end{aligned} \]
\(\frac{2}{7}\) is the area under the curve alone, and \(\frac{3}{7}\) doubles the answer as if the region were reflected into \(x \lt 0\), where the curve does not exist.
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