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When a portion of the curve $y = f(x)$ lies below the $x$-axis between $x = a$ and $x = b$, the area enclosed there is found by taking which of the following?
Athe square of the integral
Bthe negative of the curve
Cthe absolute value of the integral
Dtwice the value of the integral
Answer & Solution
Correct answer: C. the absolute value of the integral
1. Below the $x$-axis $f(x) < 0$, so $\int_a^b f(x)\,dx$ comes out negative.
2. Area is a physical magnitude and cannot be negative.
3. Hence we report its numerical value, $\left|\int_a^b f(x)\,dx\right|$.
_Source: NCERT Class 12 Mathematics Ch 8 "Application of Integrals", p.1_
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