The function \(f\) is defined by \( f(x)=\begin{cases} 5 & \text{for } x\lt 4 \\ x+1 & \text{for } x \geq 4 \end{cases} \). What is the value of \(\displaystyle \int\limits_0^6 f(x)   dx \) ?

Summary Submit Skip Question

We can solve split the integral into two parts corresponding to the two sides of the piecewise function.

$$ \int \limits_0^6 f(x)   dx = \int\limits_0^4 f(x)   dx + \int\limits_4^6 f(x)   dx $$ $$ =\int\limits_0^4 5   dx + \int\limits_4^6 x+1   dx $$ $$ = \Big[ 5x \Big]_0^4 + \Big[ \frac{1}{2}x^2+x \Big]_4^6 $$ $$ = 20-0 + 24-12 $$ $$ = \boxed{32} $$