(a)[3]
Apply the trapezium rule with three intervals to estimate $\int_{1}^{4} \frac{6}{1 + \sqrt{x}} \, dx$. Give the result correct to $5$ significant figures.
(b)[3]
Determine the exact value of $\int_{1}^{4} 2e^{\frac{1}{2}x - 2} \, dx$.
(c)[2]
The diagram presents the curves $y = \frac{6}{1 + \sqrt{x}}$ and $y = 2e^{\frac{1}{2}x-2}$, which intersect at a point whose $x$-coordinate is 4. The shaded part is enclosed by the two curves and the line $x = 1$. Use your answers to parts (a) and (b) to estimate the area of the shaded region. Give your answer correct to 3 significant figures.
(d)[1]
State, with a reason, whether your answer to part (c) is an over-estimate or under-estimate of the exact area of the shaded region.