Integral Calculus
Periodic piecewise function; definite integral
MMTS_Full_Test_10
Grade 12
Question:
Let $f$ be a function defined by $f(x)=\begin{cases}(x-r)^2, & r-1\leq x<r+1\\ 1, & r+1\leq x\leq r+2\end{cases}$ where $r=3k$, $k\in I$. Find $\displaystyle\sqrt{\int_0^{45}f(x)\,dx}$
Step-by-Step Solution
Key Concept: $f$ is periodic with period 3. $\int_0^3 f(x)dx=\int_{-1}^1(x-0)^2dx+\int_1^2 1\,dx=2/3+1=5/3$. Total $=15\cdot5/3=25$. $\sqrt{25}=5$.
$\sqrt{25}=5$.
Correct Answer: (C) 5