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 $\displaystyle \int\limits_0^6 f(x)   dx$ ?

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

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