The area of the region bounded by the line \( x=9 \) and the parabola \( y^{2}=25x \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 180 sq. units
The line x = 9 meets the parabola at \( y=\pm 15 \). The region is symmetric about the x-axis, so the area is twice the upper half, where \( y=5\sqrt{x} \):
\( \displaystyle \text{Area}=2\int_{0}^{9}5\sqrt{x}\,dx \)
\( \displaystyle =10\cdot\frac{2}{3}\left[x^{3/2}\right]_{0}^{9} \)
\( \displaystyle =10\cdot\frac{2}{3}\cdot 27 \)
\( \displaystyle =180 \)
Shortcut: A parabolic segment has \( \frac{2}{3} \) of the area of its enclosing rectangle. Here the chord is 30 long and the depth is 9:
\( \displaystyle \frac{2}{3}(30\times 9)=180 \)