If f(x) = x2 , g(x) = 2x,0 ≤ x ≤ 2 then the value of I(x) = ∫ limits02 max (f(x), g(x)) is

Question

If f(x) = x2 , g(x) = 2x,0 ≤ x ≤ 2 then the value of I(x) = ∫ limits02 max (f(x), g(x)) is (A) (8/3) (B) (2/3) (C) (5/3) (D) (7/3)

Tardigrade Admin · Accepted Answer

Let r(x)= max (f(x), g(x)) ∴ r(x)=2 x x ∈(0,1) and r(x)=x2 x ∈(1,2) ∴ I ( x )=∫ limits01 2 xdx +∫ limits12 x 2 dx ∴ I ( x )=[ x 2]01+([ x 3]12/3) =(1-0)+(8-1/3)=(7/3)

Q. If $f (x) = x^{2}, g (x) = 2 x, 0 \leq x \leq 2$ then the value of $I (x) = 0 \int 2 ma x (f (x), g (x))$ is

$\frac{8}{3}$

$\frac{2}{3}$

$\frac{5}{3}$

$\frac{7}{3}$

Let $r (x) = max (f (x), g (x))$
$∴ r (x) = 2 x x \in (0, 1)$
and $r (x) = x^{2} x \in (1, 2)$
$∴ I (x) = 0 \int 1 2 x d x + 1 \int 2 x^{2} d x$
$∴ I (x) = [x^{2}]_{0}^{1} + \frac{[ x ^{3} ] _{1}^{2}}{3}$
$= (1 - 0) + \frac{8 - 1}{3} = \frac{7}{3}$

Q. If f(x)=x2,g(x)=2x,0≤x≤2 then the value of I(x)=0∫2​max(f(x),g(x)) is

38​

32​

35​

37​