Answer is: option3

Solution:

Given:

• The curve \( y = f(x) \) intersects the line \( y = -3 \) at points \( (0, -3) \), \( (a, -3) \), and \( (b, -3) \).
• Between \( x = 0 \) and \( x = a \), \( f(x) > -3 \).
• Between \( x = a \) and \( x = b \), \( -3 > f(x) \).

Step-by-Step Solution:

1. Area between \( x = 0 \) and \( x = a \):
   • Here, \( f(x) \) is above \( y = -3 \), so the height of the region is \( f(x) - (-3) = f(x) + 3 \).
   • The area is:
     \( \int_0^a [f(x) - (-3)]\, dx = \int_0^a [f(x) + 3]\, dx \)
2. Area between \( x = a \) and \( x = b \):
   • Here, \( y = -3 \) is above \( f(x) \), so the height of the region is \( -3 - f(x) \).
   • The area is:
     \( \int_a^b [-3 - f(x)]\, dx \)
3. Total Area:
   • The sum of the two areas is:
     \( \int_0^a [f(x) + 3]\, dx + \int_a^b [-3 - f(x)]\, dx \)

Matching with Options:

• Option (C) matches exactly with the derived expression:
  \( \int_0^a [f(x) + 3]\, dx + \int_a^b [-3 - f(x)]\, dx \)