Problem

The diagram shows the curve with equation

$$y = 4(3x + 4)^{\frac{1}{2}} - 2x - 6$$

for values of $x$ such that $0 \leq x \leq 7$. The tangent to the curve at the point $P(7, 0)$ meets the $y$-axis at the point $Q$. Region $A$ is bounded by the curve and the two axes. Region $B$ is bounded by the curve, the line segment $PQ$ and the $y$-axis.

(a) Find the area of region $A$.

(b) Find the area of region $B$.