To determine

To find:

The area of the region bounded by the parabola y=x2, the tangent line to this parabola at 1, 1, and the x- axis

Answer

112

Explanation

1) Concept:

Find the equation of the tangent line to the parabola y=x2 at the point (1, 1).

Then find the area of the shaded region.

2) Given:

y=x2

3) Formula:

∫xn dx=xn+1n+1+C

4) Calculation:

To find the equation of the tangent line to the parabola y=x2 at the point (1, 1):

The slope a of this line is given by the derivative of the parabola y=x2 at 1

Differentiate y=x2.

y'=2x2-1

Simplify.

y'=2x

Put x=1 to get the slope a of this line.

a=y'1=2

Therefore, equation of line is y=2x+b.

To find b:

Since (1, 1) is on the line, it satisfies the equation of line.

Put x=1, y=1 in the equation of line y=2x+b

1=2(1)+b

Simplify.

1=2+b

Subtract 2 on both sides.

1-2=2+b-2

Simplify and rearrange.

b=-1

Therefore, the equation of the tangent line is y=2x-1.

The solution of 2x-1=0 is x=12

Therefore, the line intersects the x- axis at 12, 0

 To find the area of shaded region, calculate the area under the parabola y=x2 from 0 to 1, and then subtract from it the area under the line from 12 to 1

Hence, the area of the shaded region is

∫01x2dx-∫121(2x-1)dx

Solve integrals.

=x3301-x2-x121

Substitute limits.

=133-033-12-1-(12)2-12

Simplify.

=13-14

Simplify.

=112

The area of the region bounded by the parabola y=x2, the tangent line to this parabola at 1, 1, and the x- axis is 112.

Conclusion:

The area of the region bounded by the parabola ,y=x2, the tangent line to this parabola at 1, 1, and the x- axis is 112.