Find the area of the following two curves.

y= sqrt(x)
y= x-2

If you mean x = y^2, then ok.

Otherwise, the two curves do not bound an area.

It's better to integrate over y, and your integrand is just

(y+2) - y^2 for -1 <= y <= 2

http://www.wolframalpha.com/input/?i=plot+x%3Dy^2%2C+y%3Dx-2

To find the area between the two curves, we need to find the points of intersection first. Let's equate the two equations and solve for x:

sqrt(x) = x - 2

To solve this equation, we need to square both sides:

x = (x - 2)^2

Expanding the right side:

x = x^2 - 4x + 4

Subtracting x from both sides:

0 = x^2 - 5x + 4

Now, we can factorize the quadratic equation:

0 = (x - 1)(x - 4)

Setting each factor to zero:

x - 1 = 0 --> x = 1
x - 4 = 0 --> x = 4

So, the two curves intersect at x = 1 and x = 4.

To find the area between the curves, we need to integrate the difference between the upper curve (y = x - 2) and the lower curve (y = sqrt(x)) with respect to x, from x = 1 to x = 4.

To calculate this area, we can set up the integral as follows:

A = ∫[1 to 4] (x - 2 - sqrt(x)) dx

Now, we can compute this integral to find the area. (If you'd like, you could solve this integral using algebraic techniques or numerical methods like Simpson's rule or the trapezoidal rule.)