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.)