Integral Calculus
Area Under Curve
MMTS_Full_Test_12
Grade 12
Question:
Consider: I) If $f(x)$ is bounded for $x\in[a,b]$, then area bounded by $y=f(x)$, $x$-axis, $x=a$ and $x=b$ is $\int_a^b f(x)dx$. II) If $f(x)$ is bounded and differentiable on $[a,b]$, then the area between $y=f(x)$ and $y=f^{-1}(x)$ is equal to double the value of $\int_a^b|f(x)-x|dx$.
Only S-I is true
Only S-II is true
Both S-I and S-II are True
Both S-I and S-II are false
Step-by-Step Solution
Key Concept: Statement I is false (area = $\int|f(x)|dx$, not $\int f(x)dx$); Statement II: standard result about area between $f$ and $f^{-1}$
I) False (area $=\int_a^b|f(x)|dx$). II) True (reflection in $y=x$). Only S-II is true.
Correct Answer: 2