Find the area bounded by the curve y=x(2-x) and the line x=2y.
To find the area bounded by the curve y = x(2 - x) and the line x = 2y, we need to find the points of intersection between the curve and the line.
First, let's set the equations y = x(2 - x) and x = 2y equal to each other in order to find the points of intersection:
x(2 - x) = 2y
Expanding the left side of the equation, we get:
2x - x^2 = 2y
Now, let's rearrange the equation to solve for y:
2y = 2x - x^2
Dividing both sides by 2, we get:
y = x - (x^2 / 2)
Now, we have two equations:
1) y = x(2 - x)
2) y = x - (x^2 / 2)
Next, let's find the x-values of the points of intersection by setting the two equations equal to each other again:
x(2 - x) = x - (x^2 / 2)
Expanding and rearranging the equation gives us:
2x - x^2 = x - (x^2 / 2)
Multiplying everything by 2 to eliminate the fractions, we get:
4x - 2x^2 = 2x - x^2
Bringing all the terms to one side of the equation, we have:
3x^2 - 2x = 0
Factoring out x, we get:
x(3x - 2) = 0
Setting each factor equal to zero gives us two solutions:
1) x = 0
2) 3x - 2 = 0, which gives x = 2/3
Now that we have the x-values of the points of intersection, we can find the corresponding y-values by substituting these x-values into either of the original equations.
For x = 0, substituting into equation 2, we get:
y = 0 - (0^2 / 2) = 0
So the first point of intersection is (0, 0).
For x = 2/3, substituting into equation 2, we get:
y = (2/3) - ((2/3)^2 / 2) = 2/3 - 4/9 = 2/9
So the second point of intersection is (2/3, 2/9).
Now that we have the points of intersection, we can find the area bounded by the curve and the line.
To do this, we need to integrate the difference between the two functions over the interval from x = 0 to x = 2/3:
Area = ∫[0 to 2/3] [(x(2 - x)) - (x - (x^2 / 2))] dx
To evaluate this integral, we need to expand and simplify the function:
Area = ∫[0 to 2/3] (2x - x^2 - x + (x^2 / 2)) dx
Area = ∫[0 to 2/3] (x - x^2/2) dx
Now, integrate the function:
Area = [x^2 / 2 - (x^3 / 6)] evaluated from 0 to 2/3
Area = [(2/3)^2 / 2 - ((2/3)^3 / 6)] - [(0^2 / 2 - (0^3 / 6))]
Area = [(4/9) / 2 - (8/27) / 6] - [0 / 2 - 0 / 6]
Area = [4/18 - 8/162] - 0
Area = (72/162 - 8/162) = 64/162 = 32/81
Thus, the area bounded by the curve y = x(2 - x) and the line x = 2y is 32/81 square units.