Question

Find the area of the region bounded by the parabola y = x2, the tangent line to this parabola at the point (2, 4), and the x-axis.

Answer 1

you need to integrate it over (2,4)
so area=54/3 sq units

Answer 2

The area of the region bounded by the parabola y = x², the tangent line to this parabola at the point (2, 4), and the x-axis is 19.334 units².

Given to us

• parabola y = x²
• the tangent line to this parabola at the point (2, 4), and the x-axis.

Equation of tangent

Differentite the equation of parabola,

F(x) =  y = x²

F(x)' =  y' = 2x

when x= 2, m= 4

For the line to be tangent the equation should be linear,

y = 4x + c

4 = 4(2)+ c

4 = 8 + c

c = -8+4

c = -4

Thus, the tangent is y=4x-4.

The area can be found by, subtracting the area (ABC) from the area(DCB),

Area (ABC)

$Area (ABC) = \int_0^4 x^2$

       $=[\dfrac{x^3}{3}]^4_0\\\\= [(\dfrac{4^3}{3})-(\dfrac{0^3}{3})]\\\\= \dfrac{64}{3}\\\\= 21.334$

Area(DCB)

$Area(DCB) = \dfrac{1}{2} \times DC \times BC$

                   

                   $= 0.5 \times 1 \times 4\\\\= 2$

Area of the region ADB

Area of the region ADB = Area (ABC) - Area(DCB)

                                       = 21.344 - 2

                                       = 19.344 units²

Hence, the area of the region bounded by the parabola y = x², the tangent line to this parabola at the point (2, 4), and the x-axis is 19.344 units².

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