Let (f: [0, 2 ] [0,1] ) be the function defined by (f(x)= ^2 x ) and let (g: [0, 2 ] [0, ) ) be the function defined by (g(x)= x 2 -x^2 ). The value of 16 ^3 ₀^ 2 f(x) g(x) d x is

Step-by-step solution

Now I₁= ₀^ 2 f(x) g(x) d x= 1 2 ₀^ 2 g(x) d x (it is explained in previous question solution) i.e. 1 2 ₀^ 2 ( 4 )^2- (x- 4 )^2 dx Using a^2-x^2 = 1 2 (x a^2-x^2 +a^2 ⁻¹ ( x a ) )+C 1 2 [ (x- 4 ) 2 x 2 -x^2 + ^2 2 2 ⁻¹ ( x- 4 4 ) ]₀^ / 2 aligned & 1 2 [ (0+ ^3 64 )- (0+ ( - ^3 64 ) ) ] & 1 2 ^3 32 aligned Now 16 ^3 ^3 64 = 1 4 =0.25