Improper integrals and the mean value of a functionEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $I=\int_0^\infty e^{-2x}\,dx$ and $J=\int_1^\infty \frac{1}{x}\,dx$.

   (a)

   Find the value of $I$.

   [1 mark]

   • A$\frac12$
   • B$2$
   • C$1$
   • D$I$ does not converge

   (b)

   Which statement about $J$ is correct?

   [1 mark]

   • A$J=0$
   • B$J=1$
   • C$J$ does not converge
   • D$J=-1$

   (c)

   Use a limit to explain why $J$ has no finite value.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=\frac{1}{\sqrt{x}}$ for $0<x\le 4$. The function $f$ is not defined at $x=0$.

   (a)

   Why is $\int_0^4 f(x)\,dx$ an improper integral?

   [1 mark]

   • AThe upper limit of the range is not finite
   • BThe integrand is undefined at the lower limit
   • CThe integrand is negative somewhere in the range
   • D$f$ has no antiderivative

   (b)

   Find the value of $\int_0^4 f(x)\,dx$.

   [1 mark]

   • A$2$
   • B$8$
   • CThe integral does not converge
   • D$4$

   (c)

   Hence find the mean value of $f$ over the interval $0\le x\le4$.

   [2 marks]

   Total for question 2: 4 marks

3. 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 $0\le t\le 12$.

   (a)

   Find the exact mean temperature over the 12 hours.

   [3 marks]

   (b)

   Find the times at which the temperature equals the mean value found in (a), giving your answers in hours to 2 decimal places.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $I=\int_0^1 \ln x\,dx$ and $K=\int_0^\infty x e^{-x}\,dx$.

   (a)

   Show that $I$ converges and find its value. You may use the fact that $t\ln t\to0$ as $t\to0^+$.

   [6 marks]

   (b)

   Show that $K$ converges and find its value. You may use the fact that $te^{-t}\to0$ as $t\to\infty$.

   [6 marks]

   Total for question 4: 12 marks

End of questions