Calculate the area between the function vx and x^2 in the interval [0 , 2].

Question

Calculate the area between the function √x and x^2 in the interval [0,2].

Answer 1

This problem is tricky because for  0 <x <1  ,  sqrt(x) > x²  ; but for 1 < x < 2,  sqrt(x) < x² .  

 

If one takes the point of view that areas are always positive, the integration must be split into two intervals:

[ 0 ,1] and [1,2].  

 

Area =  ∫₀¹ [sqrt(x) - x² ] dx   +  ∫₁² [x² - sqrt(x) ] dx.  

 

Using the fact that the anti-derivative of sqrt(x) is  (2/3) x^3/2  and the anti-derivative of x² is (1/3) x³ , the Area can be worked out to be   10/3 -(4/3)sqrt(2).

 

Of course, if one allows areas to be negative, the answer can be obtained without splitting the interval.