If a continuous curve y = f(x) crosses the x-axis exactly once at x = c where a < c < b, the total geometrical area bounded by the curve, x-axis, x=a, and x=b is:

If a continuous curve y = f(x) crosses the x-axis exactly once at x = c where a < c < b, the total geometrical area bounded by the curve, x-axis, x=a, and x=b is:

• A. |∫(a to c) f(x) dx| + |∫(c to b) f(x) dx|
• B. ∫(a to b) f(x) dx
• C. |∫(a to b) f(x) dx|
• D. ∫(a to c) f(x) dx − ∫(c to b) f(x) dx always

Answer: A) |∫(a to c) f(x) dx| + |∫(c to b) f(x) dx|

Explanation: To avoid cancellation of positive and negative regions, the area must be split at the root x=c, taking the absolute value of each segment's integral.