Definite Integration
Integral of max Function
nta_pyq_2023_jan
Grade 12
Question:
Let $[x]$ denote the greatest integer $\leq x$. Consider the function $f(x)=\max\{x^2,1+[x]\}$. Then the value of $\displaystyle\int_0^2 f(x)\,dx$ is:
1. \dfrac{5+4\sqrt{2}}{3}
2. \dfrac{8+4\sqrt{2}}{3}
3. \dfrac{1+5\sqrt{2}}{3}
4. \dfrac{4+5\sqrt{2}}{3}
Step-by-Step Solution
Key Concept: On $[0,1)$: $1+[x]=1$, $x^2\leq1$, so $f(x)=1$. On $[1,\sqrt{2})$: $1+[x]=2$, $x^2\leq2$, so $f(x)=2$. On $[\sqrt{2},2]$: $x^2\geq2$, $f(x)=x^2$.
$\dfrac{5+4\sqrt{2}}{3}$.
Correct Answer: 1