How do I solve this integral. The second part throws me off

Question

I just started doing integration and I dont know what the curly bracket(f(x)= 4 for x<4 etc...) means:

Answer 1

Note that $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$. IN this case, take $a=0$, $b = 8$ and $c=4$. And then use the appropriate part of definition of $f(x)$ on each interval -- between $0$ and $4$ it is $4$, and between $4$ and $8$ it is $x$.

So, the integral is $\int_0^4 4 dx + \int_4^8 x dx$.

Answer 2

I dont know what the curly bracket(f(x)= 4 for x<4 etc...) means

That's called a piecewise function, because it's defined in pieces. It's typically prerequisite knowledge for calculus, so definitely make sure you understand it before moving too much farther.

The piecewise function you provided says:

$$ \text{If $x < 4$ then the value of $f(x)$ is $4$.} $$ $$ \text{If $x \ge 4$ then the value of $f(x)$ is $x$.} $$ For example, $f(2) = 4$ because $2 < 4$ and $f(12) = 12$ because $12 \ge 4$.

So to integrate it, split up the integral at $x=4$ as explained in Batman's answer.