Question

Assertion :${\int }_0^{10\pi }\left|\cos x\right|dx=20$ Reason: ${\int }_a^{b}f\left(x\right)dx\ge 0$

then $f\left(x\right)\ge 0\forall xϵ(a,b)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

${\int }_0^{10\pi }|cosx|dx={\int }_0^{\pi /2}20\times cosx=20[sinx{]}_0^{\pi /2}dx=20(1-0)=20$.

Reason is false because if we take $f\left(x\right)=-x$ for $a=2$ and $b=1$,

then ${\int }_2^1(-x)dx=3/2$.

hence, ${\int }_a^{b}f\left(x\right)dx>0$ but $f\left(x\right)<0.$