(a) Evaluate the definite integral by interpreting it in terms of signed area.
$\displaystyle \int_{1}^{6} (3 x + 6) \, dx =$
Suggestion: Draw a picture of the region whose signed area is represented by the integral. Then find the signed area using formulas from basic geometry.

(b) Find the average value of $f(x) = 3 x + 6$ over the interval $\lbrack 1, 6 \rbrack$.
Average value =