Improper integrals and the mean value of a functionEdexcel A-Level Further Maths: Mind map

Exam questions on Improper integrals and the mean value of a function

1. Let $I=\int_0^\infty e^{-2x}\,dx$ and $J=\int_1^\infty \frac{1}{x}\,dx$.
   Use a limit to explain why $J$ has no finite value.2 marks
2. Let $f(x)=\frac{1}{\sqrt{x}}$ for $0<x\le 4$. The function $f$ is not defined at $x=0$.
   Hence find the mean value of $f$ over the interval $0\le x\le4$.2 marks
3. The temperature, $T\,^\circ$C, in a greenhouse $t$ hours after 06:00 is modelled by $T=15+4\sin\left(\frac{\pi t}{12}\right)$ for $0\le t\le 12$.
   Find the exact mean temperature over the 12 hours.3 marks