Question

The area (in sq. units) of the region bounded by the curve \(y = 2x-x^2\) and the X-axis is...

• A. \(\frac{4}{3}\)
• B. \(\frac{8}{3}\)
• C. \(\frac{20}{3}\)
• D. \(\frac{2}{3}\)

Hint:

Find where the parabola meets the X axis and integrate between the roots.

Answer 1

The Correct Option is A

Step 1: Formula for a parabola segment
The area between a parabola $y = ax^2 + bx + c$ and the X-axis is $\dfrac{|a|(x_2 - x_1)^3}{6}$ where $x_1, x_2$ are the roots.

Step 2: Apply
Here $|a| = 1$ and the roots are 0 and 2, so the area is $\dfrac{2^3}{6} = \dfrac{8}{6}$.

Step 3: Simplify
$\frac{8}{6} = \frac{4}{3}$.

Step 4: Verify
The direct integral $\int_0^2(2x - x^2)dx = 4 - \frac{8}{3} = \frac{4}{3}$ gives the same answer.

Final Answer:
The area is 4/3 square units. This is option (A). \[ \boxed{\text{(A) }\frac{4}{3}} \]