Question

The area of the region bounded by the curve y = 2x – x² and X-axis is ______.

विकल्प

• `2/3` sq units

• `4/3` sq units

• `5/3` sq units

• `8/3` sq units

Answer 1

The area of the region bounded by the curve y = 2x – x² and X-axis is `underlinebb(4/3 sq  units)`.

Explanation:

The equation of given curve is y = 2x – x²

`\implies` x² – 2x = – y

`\implies` x² – 2x + 1 = – y + 1

`\implies` (x – 1)² = – (y – 1)

Which is the equation of parabola whose vertex is (1, 1) and it is open downward.

For intersection of the parabola with the X-axis, put y = 0, then we get

0 = 2x – x²

`\implies` x(2 – x) = 0

`\implies` x = 0, 2

Hence, area of bounded region between the curve and X-axis

= `int_0^2 ydx`

= `int_0^2 (2x - x^2)dx`

= `[(2x^2)/2 - x^3/3]_0^2`

= `[4 - 8/3 - 0 - 0]`

= `4/3` sq units.