Find the area of the region bounded by the given curves: y= `sqrt(x)` , y=x^2 and x=2.  

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The area of the region between the curves y = `sqrt(x)` , y=x^2 and x=2 has to be determined.

The curves y = `sqrt x` and y = x^2 intersect at the point (1, 1).

x = 2 and y = `sqrt x` intersects at (2, sqrt 2)

x = 2 and y = x^2 intersects at (2, 4)

The required area is the integral `int_(1)^2(x^2 - sqrt x) dx`

=> `x^3/3 - (2/3)*x^(3/2)` between 1 and 2

=> `2^3/3 - (2/3)*2^(3/2) - (1^3/3 - (2/3)*1^(3/2))`

=> `8/3 - (2/3)*2^(3/2) - 1/3 + 2/3`

The area between the three regions is `3 - (2/3)*2^(3/2)`

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