Let f ( x ) = Max { x 2 , ( 1 - x ) 2 , 2 x ( 1 - x ) } where 0 x 1 , if the area of the region bounded by the curves y = f ( x ) , x -axis, x = 0 and x = 1 is p q then p + q = ____ (H.C.F of p , q is 1)

Question

Let $f (x) = Max {x^{2}, {(1 - x)}^{2}, 2 x (1 - x)}$ where $0 \leq x \leq 1$ , if the area of the region bounded by the curves $y = f (x)$ , $x$ -axis, $x = 0$ and $x = 1$ is $\frac{p}{q}$ then $p + q =$ ____ (H.C.F of $p, q$ is 1)

Smart Achievers · Accepted Answer

(44)

From fig it clear that $f (x) =$ ${\begin{matrix} {(1 - x)}^{2}, & 0 \leq x \leq \frac{1}{3} \\ 2 x (1 - x), & \frac{1}{3} < x \leq \frac{2}{3} \\ x^{2}, & \frac{2}{3} < x \leq 1 \end{matrix}$

The required area $A = \int_{0}^{1} f (x)$ $= \int_{0}^{\frac{1}{3}} {(1 - x)}^{2} d x + \int_{\frac{1}{3}}^{\frac{2}{3}} 2 x (1 - x) d x + \int_{\frac{2}{3}}^{1} x^{2} d x$

$= {[- \frac{1}{3} {(1 - x)}^{3}]}_{0}^{\frac{1}{3}} + {[x^{2} - \frac{2 x^{3}}{3}]}_{\frac{1}{3}}^{\frac{2}{3}} +$ ${[\frac{x^{3}}{3}]}_{\frac{2}{3}}^{1}$ $= \frac{17}{27}$

So, $\frac{p}{q} = \frac{17}{27}$

Hence, $p + q = 17 + 27 = 44$

Solution

Q.
Let $f (x) = Max {x^{2}, {(1 - x)}^{2}, 2 x (1 - x)}$ where $0 \leq x \leq 1$ , if the area of the region bounded by the curves $y = f (x)$ , $x$ -axis, $x = 0$ and $x = 1$ is $\frac{p}{q}$ then $p + q =$ ____ (H.C.F of $p, q$ is 1)

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let f(x)=Max{x2,(1-x)2,2x(1-x)} where 0≤x≤1, if the area of the region bounded by the curves y=f(x), x-axis, x=0 and x=1 is pq then p+q=____ (H.C.F of p,q is 1)

Ans. (44) From fig it clear that f(x)={(1-x)2,0≤x≤132x(1-x),13<x≤23x2,23<x≤1 The required area A=∫01f(x)=∫013(1-x)2dx+∫13232x(1-x)dx+∫231x2 dx =[-13(1-x)3]013+[x2-2x33]1323+[x33]231=1727 So, pq=1727 Hence, p+q=17+27=44

Want Us to Email you About Special Offers & Updates?

Q.
Let $f (x) = Max {x^{2}, {(1 - x)}^{2}, 2 x (1 - x)}$ where $0 \leq x \leq 1$ , if the area of the region bounded by the curves $y = f (x)$ , $x$ -axis, $x = 0$ and $x = 1$ is $\frac{p}{q}$ then $p + q =$ ____ (H.C.F of $p, q$ is 1)