Question

The area bounded by the curve y = 4x − x² and the x-axis is __________ .

Options

• \[\frac{30}{7}\]sq. units

• \[\frac{31}{7}\]sq. units

• \[\frac{32}{3}\]sq. units

• \[\frac{34}{3}\]sq. units

Answer 1

\[\frac{32}{3}\]sq. units

Point of intersection of parabola y = 4x − x² with x-axis is given by 
\[y = 4x - x^2\text{ and }y = 0 ................\left(\text{Equation of x axis }\right)\]
\[ \Rightarrow 4x - x^2 = 0\]
\[ \Rightarrow x = 0\text{ or }x = 4 \]
\[ \Rightarrow y = 0 , y = 0\]
\[\text{ Thus O }\left( 0, 0 \right)\text{ and B }\left( 4, 0 \right) \text{ are points of intersection of parabola and x - axis . }\]
\[\text{ Required shaded area }= \int_0^4 \left( 4x - x^2 \right) dx\]
\[ = \left[ 2 x^2 - \frac{x^3}{3} \right]_0^4 \]
\[ = 2 \times 16 - \frac{64}{3} - 0\]
\[ = \frac{96 - 64}{3}\]
\[ = \frac{32}{3}\text{  square units }\]