Question

The area of the region bounded by \( y = x^{5/2} \) and \( y = x \) (in square units) is

• A. \( \frac{3}{7} \)
• B. \( \frac{2}{7} \)
• C. \( \frac{3}{14} \)
• D. \( \frac{5}{14} \)
• E. \( \frac{4}{7} \)

Hint:

Always check which curve is above before integrating.

Answer 1

The Correct Option is C

Concept: Area between curves: \[ \int (upper - lower)\,dx \]

Step 1: Find intersection.
\[ x^{5/2} = x \Rightarrow x=0,1 \]

Step 2: Set integral.
\[ \int_0^1 (x - x^{5/2})dx \]

Step 3: Evaluate.
\[ = \left[\frac{x^2}{2} - \frac{2}{7}x^{7/2}\right]_0^1 = \frac{1}{2} - \frac{2}{7} \] \[ = \frac{7-4}{14} = \frac{3}{14} \]