Find the area of the following two curves.
y= sqrt(x)
y= x-2
To find the area between two curves, we need to find the points of intersection where the curves intersect. In this case, we have the following two curves:
1. y = sqrt(x)
2. y = x - 2
To find the points of intersection, we set the two equations equal to each other 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)(x - 2)
x = x^2 - 4x + 4
Rearranging the terms:
x^2 - 5x + 4 = 0
Factoring the quadratic equation:
(x - 1)(x - 4) = 0
Setting each factor equal to zero:
x - 1 = 0 or x - 4 = 0
Solving for x:
x = 1 or x = 4
Now we have the x-values of the points of intersection: x = 1 and x = 4. We can substitute these values back into either equation to find the corresponding y-values.
For y = sqrt(x):
When x = 1:
y = sqrt(1) = 1
When x = 4:
y = sqrt(4) = 2
So the points of intersection are (1, 1) and (4, 2).
To find the area between the curves, we need to subtract the bottom curve from the top curve and integrate it over the interval [1, 4]. In this case, the top curve is y = x - 2 and the bottom curve is y = sqrt(x).
The area between the curves can be found using the definite integral:
A = ∫[1,4] (x - 2 - sqrt(x)) dx
Evaluating this integral will give us the area between the curves.