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

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KollegeApply · Accepted Answer

$\frac{32}{3}$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 - x^$\frac{3}{3}$ \right]_0^4 = 2 $\times$ 16 - $\frac{64}{3}$ - 0 = 96 - $\frac{64}{3}$ = $\frac{32}{3}$\text{ square units } [Solution diagram: 44111_solution_1.png]

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