Final answer:
To find the area of the region bounded by the parabola y = 4x², the tangent line to the parabola at (2, 16), and the x-axis, we need to find the x-coordinates of the points where the parabola and the tangent line intersect.
Step-by-step explanation:
To find the area of the region bounded by the parabola y = 4x², the tangent line to the parabola at (2, 16), and the x-axis, we need to find the x-coordinates of the points where the parabola and the tangent line intersect.
Since the tangent line goes through the point (2, 16), we can find its equation using the point-slope form, which is y - y1 = m(x - x1).
Substituting the values, we get y - 16 = m(x - 2), and since the line is tangent to the parabola, their slopes will be equal. So, the slope of the tangent line is equal to the derivative of the parabola at x = 2.