Find the area of the region bounded by the parabola y=3x^2

Question

the tangent line to this parabola at (2,12) and the x axis.?

Answer 1

The parabola is y = 3x²

Apply derivative on each side with respect of x.

y' = 6x

Substitute x = 2 in y'.

y' = 6(2)

y'= 12

This is the slope of tangent line to the curve at (2, 12).

To find the tangent line equation substitute m = 12 and (x ,y) = (2, 12) in y = mx + b.

12 = 12(2) + b

b = 12 - 24

b = - 12

Substitute m = 12 and b = - 12 in y = mx + b.

y = 12x - 12

Answer 2

Contd...

The region bounded by the parabola y = 3x² and the line y = 12x - 12 and x - axis equation is  y = 0

12x - 12 = 0

12x = 12

x = 1

Tangent line meets the x axis at 1.

To find the area we need to integral

In this case f(x) = 3x², g(x) = 12x - 12

f(x) > g(x)

Area is 7 square units.