Find the area of the following two curves.

y= sqrt(x)
y= x-2

Note: People have been telling me that the two lines do not intersect but they do there has to be an answer. please help me.
Thanks

The lines do intersect.

sqrt(x) = x-2
x =(x-2)^2 =x^2-4x + 4
x^2-5x+4=0
(x-1)(x-4)=0
x=1,4 but 1 does not work if you plug in x=1 to both equations, so the only answer is x=4.

Jim, they do intersect at one point only, thus creating an open region on the left.

By squaring you are introducing a part of the graph which was originally undefined.

y = √x is defined only for x ≥ 0

Mia:
Both Steve and I gave answers to this earlier
http://www.jiskha.com/display.cgi?id=1398476046

Are you finding the area between the two curves from the y-axis to the point (4,2) ?

That looks straightforward and should pose no problems for you.
However, you did not say that.

To find the area between the curves, we need to determine the points of intersection between the two curves.

Let's set the two equations equal to each other:
sqrt(x) = x - 2

To simplify this equation, we'll 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 solve this quadratic equation by factoring or using the quadratic formula. Factoring, we have:
0 = (x - 4)(x - 1)

Setting each factor equal to zero:
x - 4 = 0 and x - 1 = 0

This gives us two solutions: x = 4 and x = 1.

To determine the y-values at these intersection points, we can substitute the x-values into either of the original equations.

For the first curve, y = sqrt(x):
For x = 4, y = sqrt(4) = 2
For x = 1, y = sqrt(1) = 1

For the second curve, y = x - 2:
For x = 4, y = 4 - 2 = 2
For x = 1, y = 1 - 2 = -1

From the given information, it seems that there is an error in the statement that the two curves do not intersect, as we have found that they intersect at the point (4, 2) and (1, -1).

To find the area between the curves, we need to determine which curve is on top in each interval. Let's analyze the intervals separately:

Interval 1: x ∈ [1, 4]
In this interval, the curve y = sqrt(x) lies above the curve y = x - 2.

Interval 2: x ∈ [4, ∞)
In this interval, the curve y = x - 2 lies above the curve y = sqrt(x).

Now, to find the area between the two curves, we integrate the difference between the curves over the intervals:

Area = ∫[1,4] [sqrt(x) - (x - 2)] dx + ∫[4, ∞) [(x - 2) - sqrt(x)] dx

Evaluating these integrals will give us the area between the curves.

Please note that it's always a good idea to double-check the given information and calculations to ensure accuracy.