Question

The area between the parabola \(y=2x-x^{2}\) and the X-axis is .....

• A. \(\frac{2}{3}\)
• B. \(\frac{4}{3}\)
• C. \(\frac{8}{3}\)
• D. \(\frac{16}{3}\)

Hint:

Find where the parabola meets the X-axis and integrate between those points.

Answer 1

The Correct Option is B

Step 1: Use the Parabola Area Formula
For a parabola cutting the X-axis at two points a distance $d$ apart with leading coefficient $|k|$, the area is $\dfrac{k d^{3}}{6}$.

Step 2: Substitute
Here $k=1$ and $d=2$.

Step 3: Compute
Area $=\dfrac{1\times8}{6}=\dfrac{4}{3}$.

Final Answer:
The area equals 4/3. \[ \boxed{\text{(B) }\dfrac{4}{3}} \]