Question

The area of the region bounded by the curve \( y = 4x - x^2 \) and the x-axis is

• A. \( \frac{16}{3} \) sq. units
• B. \( \frac{32}{3} \) sq. units
• C. 32 sq. units
• D. 16 sq. units

Hint:

To find the area under a curve, integrate the function between the x-intercepts (or other relevant boundaries).

Answer 1

The Correct Option is B

Step 1: Set up the integral for the area.
To find the area, we need to integrate the given function from 0 to the point where it intersects the x-axis. First, solve for the x-intercepts of the curve by setting \( y = 0 \). \[ 0 = 4x - x^2 \quad \Rightarrow \quad x(4 - x) = 0 \] Thus, the x-intercepts are at \( x = 0 \) and \( x = 4 \).
Step 2: Integrate the function.
Now, integrate the function \( y = 4x - x^2 \) from 0 to 4: \[ \text{Area} = \int_0^4 (4x - x^2) \, dx \] Evaluating the integral, we get the area as \( \frac{32}{3} \) square units.
Step 3: Conclusion.
Thus, the area of the region is \( \frac{32}{3} \), corresponding to option (B).