Final answer:
The area bounded by the curves y=√6x and y= x²/6 is given by 3√6 - 12.
Step-by-step explanation:
To find the area of the plane figure bounded by the curves y=√6x and y= x²/6, we need to determine the points of intersection between these two curves. Setting them equal to each other, we have √6x = x²/6. Simplifying, we get x³-6√6x=0, which factors as x(x-√6)(x+√6)=0. This gives us three x-values: x=0, x=√6, x=-√6. We can then determine the corresponding y-values by substituting these x-values into the equations. Plugging x=0 into both equations, we get y=0. Plugging x=√6 into the first equation, we get y=6. Plugging x=-√6 into the first equation, we get y=-6.
Next, we can calculate the area between the curves by integrating the difference between the two equations with respect to x from x=-√6 to x=√6. The area is given by the integral of (√6x - x²/6) dx. Evaluating this integral, we get [(√6/2)x² - (1/18)x³] evaluated from x=-√6 to x=√6. Plugging in the values, we get [(√6/2)(6)-(1/18)(6³)]-[(√6/2)(-6)-(1/18)(-6³)]. Simplifying, we get (3√6)-(1/18)(216) = 3√6 - 12 = -12 + 3√6.