Find the area of the shaded region.
y= x-2
y= sqrt(x)
i need it in 8 mins
To find the area of the shaded region between the two curves, we need to determine the points of intersection and then integrate the difference between the two curves over that interval.
Step 1: Find the points of intersection.
Set the equations of the two curves equal to each other:
x - 2 = √x
Square both sides of the equation to eliminate the square root:
(x - 2)^2 = x
Expand the equation:
x^2 - 4x + 4 = x
Rearrange to get a quadratic equation:
x^2 - 5x + 4 = 0
Factorize the equation:
(x - 4)(x - 1) = 0
Set each factor to zero and solve for x:
x - 4 = 0 -> x = 4
x - 1 = 0 -> x = 1
Step 2: Determine the interval of integration.
The shaded area lies between x = 1 and x = 4.
Step 3: Setup and evaluate the integral.
The area can be calculated by integrating the difference of the two curves over the interval [1, 4]:
Area = ∫(sqrt(x) - (x - 2)) dx from 1 to 4
The integral can be evaluated to find the area of the shaded region.