Show that $5\cos 2t-5\cos 6t\approx 80t^2$ for small values of $t$.

The output signal of a high-frequency sensor is modeled by the function $R(t)=\sqrt{\frac{1}{20}(5\cos 2t-5\cos 6t)}$. Show that for small positive values of $t$, the area enclosed by the curve $y=R(t)$, the $t$-axis, and the line $t=0.4$ can be approximated by $Area\approx 2^m\times 5^n$ where $m$ and $n$ are integers to be found.

Explain why ${\int }_{12.6}^{12.7}2tdt$ is not a suitable approximation for ${\int }_{12.6}^{12.7}R(t)dt$.

Explain how ${\int }_{12.6}^{12.7}R(t)dt$ may be approximated by ${\int }_a^{b}2tdt$ for suitable values of $a$ and $b$.