How may I integrate $I = \int_{0}^{a} f(x) x^2 dx $?

Question

How may I integrate $I = \int_{0}^{10} f(x) x^2 dx $ where $f(x)$ is:

$$ \begin{equation} f(x)=\begin{cases} f_1, & x \in (0,8) ∪ (9,10)\\ f_2, & x \in (8,9) \end{cases} \end{equation} $$

EDIT: upper interval is 10, not $a$

Answer 1

Assuming that $f_1, f_2$ are constants, just consider all possible values for $a$.

1. If $a < 8$, $$\int_0^a f(x) x^2 dx = \int_0^a f_1 x^2 dx = \frac{f_1 a^3}{3}.$$

2. If $8 \leq a \leq 9$, $$ \int_0^2 f(x) x^2 dx = \int_0^8 f_1 x^2 dx + \int_8^a f_2 x^2 dx = \frac{f_1 8^3}{3} + \left(\frac{f_2 a^3}{3}-\frac{f_2 8^3}{3}\right) = \frac 13 f_2 a^3+\left(f_1-f_2\right)\frac{512}{3} $$

I trust you can now deal with the case $a>9$.