Consider the curves f(x)=x|x|-1 and g(x)= cases 3 x 2 , & x 0 2 x, & x 0 cases What is the area bounded by the curves?

Step-by-step solution

aligned Area & = _ -1 ^2[g(x)-f(x)] d x & = _ -1 ^0[g(x)-f(x)] d x+ ₀^2[g(x)-f(x)] d x & = _ -1 ^0 (2 x+x^2+1 ) d x+ ₀^2 [ 3 x 2 -x^2+1 ] d x & =I₁+I₂ I₁ & = ( 2 x^2 2 + 1 3 x^3+x )_ -1 ^0 aligned aligned & =(0+0+0)- ((-1)^2+ 1 3 (-1)^3+(-1) ) & =- (1- 1 3 -1 )= 1 3 I₂ & = ₀^2 [ 3 x 2 -x^2+1 ] d x & = [ 3 2 x^2 2 - 1 3 x^3+x ]₀^2 & = ( 3 4 4- 1 3 8+2 )-(0) & =5- 8 3 = 7 3 Area & =I₁+I₂= 1 3 + 7 3 aligned Area = 8 3 square units.