Describe how the leading-order approximation may be found by the method of stationary phase of

$$I(\lambda)=\int_{a}^{b} f(t) \exp (i \lambda g(t)) d t$$

for $\lambda \gg 1$, where $\lambda, f$ and $g$ are real. You should consider the cases for which: (a) $g^{\prime}(t)$ has one simple zero at $t=t_{0}$, where $a<t_{0}<b$; (b) $g^{\prime}(t)$ has more than one simple zero in the region $a<t<b$; and (c) $g^{\prime}(t)$ has only a simple zero at $t=b$.

What is the order of magnitude of $I(\lambda)$ if $g^{\prime}(t)$ is non zero for $a \leqslant t \leqslant b$ ?

Use the method of stationary phase to find the leading-order approximation for $\lambda \gg 1$ to

$$J(\lambda)=\int_{0}^{1} \sin \left(\lambda\left(t^{3}-t\right)\right) d t$$

[Hint:

$$\left.\int_{-\infty}^{\infty} \exp \left(i u^{2}\right) d u=\sqrt{\pi} e^{i \pi / 4} .\right]$$