Final answer:
The area of the plane figure bounded by the curves y = 12 - 3x², y = 0, and x = 0 is 16 square units, calculated by integrating the function from 0 to 2.
Step-by-step explanation:
To find the area of the plane figure bounded by the curves y = 12 - 3x², y = 0, and x = 0, we need to set up an integral. Since the curve y = 12 - 3x² is a parabola that opens downwards with a vertex at (0, 12), the intersection with y = 0 will give us the limits of integration for x. Setting y to 0, we solve for x to find the points where the parabola intersects the x-axis: 0 = 12 - 3x², which simplifies to x² = 4, so x = -2 and x = 2. However, since we're only looking at x = 0 and x greater than 0, we ignore the negative solution.
The area can be calculated as the integral from 0 to 2 of (12 - 3x²) dx. Working through the integral, we get:
∫ (12 - 3x²) dx = [12x - x³] evaluated from 0 to 2 = (12*2 - 2³) - (12*0 - 0³) = 24 - 8 = 16 square units.