Question

If $f\left(x\right)=\{\begin{array}{c}{x}^{2}0\le x\le 1\\ \sqrt{x}1\le x\le 2\end{array}$, Integrate $f\left(x\right)$

• A. $\frac{2}{3}(2\sqrt{2}-1)$
• B. $\frac{1}{3}(4\sqrt{2}+1)$
• C. $\frac{1}{3}(4\sqrt{2}-1)$
• D. $\frac{1}{2}(4\sqrt{2}-1)$

Answer 1

The correct option is C $\frac{1}{3}(4\sqrt{2}-1)$

$f\left(x\right)=\{\begin{array}{c}{x}^{2}0\le x\le 1\\ \sqrt{x}1\le x\le 2\end{array}$

$\therefore {\int }_0^{2}f\left(x\right)dx={\int }_0^1{x}^{2}dx+{\int }_1^2\sqrt{x}dx$

$={\left[\frac{x^3}{3}\right]}_0^1+{\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]}_1^2=\frac{1}{3}(4\sqrt{2}-1)$