Determine the exact value for the constant k such that the area of the region bounded by the curves y=x and y=kx^2 is equal to 2/3 sq units.
Any help is appreciated.
To find the value of the constant k that satisfies the given condition, we need to find the points of intersection between the curves y = x and y = kx^2 and then calculate the area between these two curves.
Step 1: Find the points of intersection
Set the two equations equal to each other:
x = kx^2
Rearrange the equation:
kx^2 - x = 0
Factor out x:
x(kx - 1) = 0
Set each factor equal to zero:
x = 0, or kx - 1 = 0
For the first equation, x = 0, the second equation, kx - 1 = 0, gives:
k = 1/x
Note that we need to find the values of x where kx - 1 as well as y = x to determine the region of intersection.
Step 2: Calculate the area
To find the area between the curves, we need to integrate the difference between the upper curve (kx^2) and the lower curve (x) over the interval of x values where they intersect.
Integrate the difference between the two curves over this interval:
Area = ∫[a, b] (kx^2 - x) dx
where a and b are the x-values where the curves intersect.
Step 3: Solve for the constant k
Given that the area is equal to 2/3 sq units, we can set up the following equation:
2/3 = ∫[a, b] (kx^2 - x) dx
Solving this equation will give us the value of the constant k that satisfies the condition.
At this point, you can proceed by calculating the integral and solving for k numerically using numerical methods or by applying definite integration techniques.