What is the exact area bounded by the parabola y = 2x^2, the y-axis, and the tangent line at (1, 2)?

To find the area bounded by the parabola ( y = 2x^2 ), the y-axis, and the tangent line at the point ( (1, 2) ), we will first need to determine the equation of the tangent line at that point.

1. Find the slope of the tangent line:

The derivative of ( y = 2x^2 ) gives us the slope of the tangent line. The derivative is ( y' = 4x ). At ( x = 1 ):

[

y' = 4(1) = 4

]

Therefore, the slope of the tangent line at ( (1, 2) ) is 4.

2. Equation of the tangent line:

We can use the point-slope form of the line equation to find the tangent line at ( (1, 2) ):

[

y - 2 = 4(x - 1)

]

Simplifying this:

[

y - 2 = 4x - 4 \implies y = 4x - 2

]

3. **Finding the