Question

If \( (a, b) \) is the stationary point of the curve \( y = 2x - x^2 \), then the area bounded by the curves \( y = 2x - x^2 \), \( y = x^2 - 2x \), and \( x = a \) is:

• A. \( \frac{3 \log 2 + 4}{2} \)
• B. \( \frac{3 + \log 4}{6} \)
• C. \( \frac{3 - \log 4}{3} \)
• D. \( \frac{1}{\log^2 2} + \frac{3}{4} \)

Hint:

For problems involving areas between curves, find the points of intersection first and then set up the definite integrals for the area calculation.

Answer 1

The Correct Option is C

We are given the curve \( y = 2x - x^2 \) and we need to find the area bounded by the curves \( y = 2x - x^2 \), \( y = x^2 - 2x \), and \( x = a \). 

Step 1: Find the stationary point by differentiating the curve \( y = 2x - x^2 \): \[ \frac{dy}{dx} = 2 - 2x. \] Setting this equal to zero, we find \( x = 1 \), so \( a = 1 \). 

Step 2: Calculate the area between the curves by setting up the definite integral between the points where the curves intersect. 

Step 3: Using the standard integration techniques and solving the definite integrals, we obtain the area as \( \frac{3 - \log 4}{3} \). Thus, the correct answer is \( \frac{3 - \log 4}{3} \).