Question

Assertion :$\sum _{r-D}^{n-1}\frac{1}{n}(\sqrt{\frac{r}{n}}+1)<{\int }_0^1(\sqrt{x}+1)dx<\sum _{r=1}^n\frac{1}{n}(\sqrt{\frac{r}{n}}+1),nϵN.$ Reason: If f (x) is continuous and increasing in [0,1], then $\sum _{r=0}^{n-1}\frac{1}{n}f\left(\frac{r}{n}\right)<{\int }_0^{1}f\left(x\right)dx<\sum _{r=1}^n\frac{1}{n}f\left(\frac{r}{n}\right)$ where n $ϵN$

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

${\sum }_{r=0}^{r=n-1}\frac{1}{n}(\sqrt{\frac{r}{n}}+1)={\int }_0^{b\cong 1}(\sqrt{x}+1)dx<{\int }_0^1(\sqrt{x}+1)dx{\sum }_{r=1}^{r=n}\frac{1}{n}(\sqrt{\frac{r}{n}}+1)={\int }_{a\cong 0}^1(\sqrt{x}+1)dx>{\int }_0^1(\sqrt{x}+1)dx$