Question

Area of the region bounded by the curve y² = 4x, y-axis and the line y = 3, is

विकल्प

• 2

• \[\frac{9}{4}\]
• \[\frac{9}{3}\]
• \[\frac{9}{2}\]

Answer 1

\[\frac{9}{4}\]

y² = 4x represents a parabola  with vertex at origin O(0, 0) and symmetric about +ve x-axis
y = 3 is a straight line parallel to the x-axis
Point of intersection of the line and the parabola is given by
Substituting y = 3  in the equation of the parabola
\[y^2 = 4x\]
\[ \Rightarrow 3^2 = 4x\]
\[ \Rightarrow x = \frac{9}{4}\]
\[\text{ Thus A }\left( \frac{9}{4} , 3 \right)\text{ is the point of intersection of the parabola and straight line }.\]
Required area is the shaded area OABO
Using the horizontal strip method ,
\[\text{ Area }\left( OABO \right) = \int_0^3 \left| x \right| dy\]
\[ = \int_0^3 \frac{y^2}{4} dy\]
\[ = \left[ \frac{1}{4}\left( \frac{y^3}{3} \right) \right]_0^3 \]
\[ = \frac{3^3}{12}\]
\[ = \frac{9}{4}\text{ sq . units }\]