Question

Given Function f(x) =$=\left\{\begin{array}{cc}{x}^2,& for0\le x<1\\ \sqrt{x},& for1\le x\le 2\end{array}\right\}$ Evaluate ${\int }_0^{2}f\left(x\right)dx$

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

Answer 1

Answer: (A)

$\frac{1}{3}(4\sqrt{2}-1)$

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