Show that the first two terms of the binomial expansion of $f(x)=\sqrt{16-8x^2}$ are $4-x^2$

State the range of values of $x$ for which the expansion found in part (a) is valid.

The signal intensity $S(\theta )$ of a specialized sensor is modeled by the function $S(\theta )=\sqrt{16\cos \theta }$ for small angles $\theta$ (in radians).

Using the result from part (a) and a suitable small angle approximation for $\cos \theta$, find an approximation for the total energy $E$ detected across a narrow sweep: $E={\int }_0^{0.4}\sqrt{16\cos \theta }d\theta$ giving your answer to four decimal places. Fully justify your answer.

A student attempts to estimate the total energy across a wider sweep using the same method: $E_{wide}={\int }_0^{1.5}\sqrt{16\cos \theta }d\theta$ Explain why this method is not suitable for this range.