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The area of the region bounded by the curve $y = f(x)$, the $x$-axis and the ordinates $x = a$ and $x = b$ (with $b > a$) is given by which expression?
A$\int_a^b f(x)\,dx$
B$\int_a^b f'(x)\,dx$
C$\int_a^b x\,dy$
D$\int_a^b \frac{1}{f(x)}\,dx$
Answer & Solution
Correct answer: A. $\int_a^b f(x)\,dx$
1. An elementary vertical strip has height $y = f(x)$ and width $dx$, so its area is $dA = y\,dx$.
2. Summing strips from $x = a$ to $x = b$ gives $A = \int_a^b y\,dx = \int_a^b f(x)\,dx$.
3. We integrate the function itself, not its derivative, so the area is $\int_a^b f(x)\,dx$.
_Source: NCERT Class 12 Mathematics Ch 8 "Application of Integrals", p.2_
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