Given that the curve y = x^3 has a tangent line that passes through point (0, 2), find the area of the region enclosed by the curve and the line by the following steps.
The Attempt at a Solution
Let f(x) = x^3 and let the coordinates of the point of tangency be (t, t^3)
f'(x) = 3x^2, f'(t) = 3t^2
The equation of the tangent line is:
y = 3t^2(x-t)+t^3
= 3t^2x - 2t^3
I am having trouble understand this part of the process. Particularly this step, y = 3t^2(x-t) + t^3. I have no idea what is happening at this step. Everything else I am completely aware of what is happening. If someone could help that would be great. Thanks!