Paragraph: 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 ). Question: The value of 2 ₀^ 2 f(x) g(x) d x- ₀^ 2 g(x) d x is

Step-by-step solution

I=2 ₀^ 2 ^2 x x 2 -x^2 _ I₁ - ₀^ 2 g(x) d x Let I₁= ₀^ 2 ^2 x ( 4 )^2- (x- 4 )^2 (making perfect square) apply kings I₁= ₀^ 2 ^2 x ( 4 )^2- ( 2 -x )^2 add both 2 I₁= ₀^ 2 ( 4 )^2- (x- 4 )^2 i.e. 2 I₁= ₀^ 2 g(x) Now I=2 I₁- ₀^ 2 g(x)=0