Question

G find the area of the region bounded by the parabola y = 2x2, the tangent line to this parabola at (1, 2), and the x-axis.

Answer 1

Please express the eqn of the parab by y = 2x^2.  " ^ " indicates exponentiation.

Find the equation of the tangent line to y = 2x^2 at (1,2):

dy/dx = 4x; now let x = 1.   Thus, the slope of the TL at (1,2) is 4(1) = 4, and the eqn of the TL is   y-2 = 4(x-1), or y-2 = 4x -4, or y = 4x -2.

To find the area bounded by the parabola, the line y = 4x-2 and the x-axis, we need to use horizontal strips and integrate with respect to y instead of to x.

The smallest y value will be 0 (this is the x-axis), and the largest will be 2 (this comes from the given point, (1,2) ).

Solve y = 2x^2 for x:  x^2 = y/2, so that x = sqrt(y/2) for y = 0 to y = 2.

                                                                        y+2
Solve y = 4x - 2 for x:  4x = y + 2, so that x = ------- 
                                                                           4

Thus, the length of each horiz. strip of width dy is given by 

(y+2)
------- - sqrt(y/2), since the first term represents the larger x-value and the 
    4                      second term represents the smaller x-value.

Then the area is the integral from y=0 to y=2 of


     (y+2)
[   --------  - sqrt(y/2)   ] dy 
        4

Can you do this integration?  If not, ask specific questions so that I could help you further.