Question

Find the area bounded by the x-axis the curve y =2x2
and the tangent to the curve at the point whose abscissa is 2

Answer 1

Answer:

The area of the region is $\dfrac{ 4}{3}$  unit square  .

Step-by-step explanation:

Given as :

The curve is y = 2 x²

The tangent to the curve at abscissa , x = 2

So, The point , y = 2 × (2)²

i.e  y = 8

The point is (2 ,8)

Now, Slop $\dfrac{\partial y}{\partial x}$

Or, $\dfrac{\partial y}{\partial x}$ = $\frac{\partial 2y^{2}}{\partial x}$

Or, slope = m = 4 y

Put the value of y

So, m = 4 × 2 = 8

So, equation of line with slope 8 and point (2 , 8)

y - 8 = 8 (x - 2)

Or, y - 8 = 8 x - 16

Or, y = 8 x - 8

It meet x-axis , so, y = 0

Or 0 = 8 x - 8

Or, 8 x = 8

So, x = 1

Point is (1 , 0)

Now, Area of shaded region equal to required area

Area = $\int_{0}^{2}$ 2 x² dx - $\int_{1}^{2}$ (8 x - 8) dx

or, 2 $\dfrac{x^{3}}{3}$ ( 0 to 2 ) - [ 8 $\dfrac{x^{2} }{2}$ - 8 x ] (1 to 2 )

Or, $\dfrac{2}{3}$ [ -0³ + 2³ ] - [ - 4 (1² - 2²) + 8 (1 - 2) ]

Or,  $\dfrac{16}{3}$ - [  12 - 8 ]

or,   $\dfrac{16}{3}$ - 4

Or, $\dfrac{ 16 -12}{3}$

Or, $\dfrac{ 4}{3}$

Hence, The area of the region is $\dfrac{ 4}{3}$  unit square  . Answer