Question

Find the area of the region bounded by the parabola y = 4x^2, the tangent line to this parabola at (2, 16), and the x-axis.

you must use integration to solve the problem and the answer cannot include variables of x and y. Please solve completely.

Answer 1

The area of region bounded by the equations is 2.66 unit²

How to calculate the area of region bounded by curve and tangent

Given equation

y = 4x²

Differentiate with respect to x

dy/dx = 8x

At point (2,16),

Slope,m is

m = 8(2) = 16

Equation of tangent is

y-y₁ = m(x-x₁).

y - 16 = 16(x-2)

y -16 = 16x - 32

y = 16x -16

Express the tangent equation and the curve in terms of x.

y = 4x²

x = √(y/4)

= √y/2

y = 16x - 16

16x = y + 16

x = (y + 16)/16

Area of region bounded by the equations

A = ∫₀¹⁶{(y + 16)/16 - y¹ᐟ²/2}

= [1/16(y²/2 + y) - 1/2(y³ᐟ²/3/2)]₀¹⁶

= [1/16(16²/2 + 16) - 1/2(16³ᐟ²/3/2) - 0]

= 8 + 16 - 64/3

= 24 - 21.33

= 2.66 unit²

Therefore, the area of region bounded by the equations is 2.66 unit²