Integral Calculus
Even periodic function; definite integral with shifted limits
Grade Class 12
Question:
If $f(x)$ is even and periodic with period $T$, $\int_0^a f(x)dx=3$ and $\int_{-T/2}^{3T/2}f(x)dx=18$, then $\int_{-a}^{a+5T}f(x)dx$ is
Step-by-Step Solution
Key Concept: From periodicity and evenness: $\int_0^T f\,dx=9$ (since $\int_{-T/2}^{3T/2}=2T\cdot\text{avg}$). $\int_{-a}^{a+5T}=\int_{-a}^0+\int_0^a+5\int_0^T=3+3+45=51$.
$\int_{-a}^{a+5T}f\,dx=3+3+45=51$.
Correct Answer: 3