Final answer:
To find the area bounded by the curves y = x√6 and y = x^2/6, we set the two equations equal to each other and solve for x. Then, we integrate the difference between the curves over the interval where they intersect, which is from x = 0 to x = 6√6. The area is given by the definite integral of (x^2/6 - x√6) with respect to x over the interval [0, 6√6].
Step-by-step explanation:
To find the area bounded by the curves y = x√6 and y = x^2/6, we need to determine the points where these curves intersect and integrate the difference between them over that interval. First, we set the two equations equal to each other and solve for x:
x√6 = x^2/6
Multiplying both sides by 6 gives us: 6x√6 = x^2
Rearranging the equation and squaring both sides, we get: x^2 - 6x√6 = 0
Factoring out an x gives us: x(x - 6√6) = 0
Therefore, x = 0 or x = 6√6. Now, we can integrate the difference between the two curves over the interval [0, 6√6] to find the area:
Area = ∫(x^2/6 - x√6) dx
Integrating the expression with respect to x, we get: (1/18)x^3 - (2√6/3)x^2
Plugging in the limits of integration, the area bounded by the curves is:
Area = ((1/18)(6√6)^3 - (2√6/3)(6√6)^2) - ((1/18)(0)^3 - (2√6/3)(0)^2)