Further calculusAQA A-Level Further Maths: Topic test

1. 1
   The region $R$ is bounded by the curve $y=\sin x$, for $0\leq x\leq\pi$, and the $x$-axis.

   (a)

   The region $R$ is rotated through $2\pi$ radians about the $x$-axis. Find the volume of the solid formed.

   [1 mark]

   • A$\frac{\pi^2}{2}$
   • B$\frac{\pi}{2}$
   • C$2\pi$
   • D$\pi^2$

   (b)

   Find the mean value of $\sin x$ over the interval $0\leq x\leq\pi$.

   [1 mark]

   • A$2$
   • B$\frac{2}{\pi}$
   • C$\frac1\pi$
   • D$\frac\pi2$

   (c)

   Find the mean value of $\sin^2x$ over the interval $0\leq x\leq\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $I=\int_0^1x^2\ln x\,\mathrm{d}x$.

   (a)

   Find the value of $I$.

   [1 mark]

   • A$\frac19$
   • B$-\frac13$
   • C$-\frac19$
   • DThe integral does not converge

   (b)

   Which statement is used when evaluating $I$ by parts?

   [1 mark]

   • A$t^3\ln t\to-\infty$ as $t\to0^+$
   • B$t^3\ln t\to0$ as $t\to\infty$
   • C$\ln t\to0$ as $t\to0^+$
   • D$t^3\ln t\to0$ as $t\to0^+$

   (c)

   Evaluate $\int_1^\infty\dfrac{1}{x(x+1)}\,\mathrm{d}x$, giving your answer in exact form.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $\mathrm{F}(x)=x\arctan x-\frac12\ln\left(1+x^2\right)$.

   (a)

   Show that $\mathrm{F}'(x)=\arctan x$.

   [3 marks]

   (b)

   Hence find the exact value of $\int_0^1\arctan x\,\mathrm{d}x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{g}(x)=\dfrac{3x^2+5x+8}{(x+2)(x^2+1)}$. The curve $C$ has equation $y=\dfrac{x^2}{4}-\dfrac12\ln x$ for $1\leq x\leq\mathrm{e}$.

   (a)

   Express $\mathrm{g}(x)$ in the form $\dfrac{A}{x+2}+\dfrac{Bx+C}{x^2+1}$, and hence find $\int_0^1\mathrm{g}(x)\,\mathrm{d}x$, giving your answer in exact form.

   [6 marks]

   (b)

   Find the exact length of the curve $C$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The region $R$ is bounded by the curve $y=\dfrac1x$, the $x$-axis and the lines $x=1$ and $x=3$. Lengths are in centimetres.

   (a)

   The region $R$ is rotated through $2\pi$ radians about the $x$-axis. Find the volume of the solid formed.

   [1 mark]

   • A$\pi\ln3\ \text{cm}^3$
   • B$\frac23\ \text{cm}^3$
   • C$\frac{4\pi}{3}\ \text{cm}^3$
   • D$\frac{2\pi}{3}\ \text{cm}^3$

   (b)

   Find the mean value of $\dfrac1x$ over the interval $1\leq x\leq3$.

   [1 mark]

   • A$\ln3$
   • B$\frac23$
   • C$\frac{\ln3}{2}$
   • D$-\frac{\ln3}{2}$

   (c)

   The line $x=3$ is replaced by the line $x=k$, where $k>1$. The volume of the solid formed when the new region is rotated through $2\pi$ radians about the $x$-axis is $\dfrac{3\pi}{4}\ \text{cm}^3$. Find $k$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $I=\int_0^{3/2}\dfrac{1}{\sqrt{9-x^2}}\,\mathrm{d}x$.

   (a)

   Find the value of $I$.

   [1 mark]

   • A$\frac\pi2$
   • B$\frac\pi6$
   • C$\frac\pi3$
   • D$\frac\pi{18}$

   (b)

   Which substitution is the most suitable for evaluating $I$?

   [1 mark]

   • A$x=3\sin\theta$
   • B$x=3\tan\theta$
   • C$x=3\sec\theta$
   • D$x=9\sin\theta$

   (c)

   Evaluate $\int_0^3\dfrac{1}{9+x^2}\,\mathrm{d}x$, giving your answer in exact form.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   For integers $n\geq0$, let $I_n=\int_0^{\pi/4}\tan^nx\,\mathrm{d}x$.

   (a)

   Show that $I_n+I_{n-2}=\dfrac{1}{n-1}$ for $n\geq2$.

   [3 marks]

   (b)

   Given that $I_0=\dfrac\pi4$, find the exact value of $I_4$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   In this question, solids are formed by rotating curves through $2\pi$ radians about the $x$-axis. Lengths are in centimetres.

   (a)

   The curve $C$ has equation $y=\frac12\left(\mathrm{e}^x+\mathrm{e}^{-x}\right)$ for $0\leq x\leq1$. Find the exact area of the curved surface formed when $C$ is rotated about the $x$-axis.

   [6 marks]

   (b)

   The curve $D$ has equation $y=\sqrt{x}\,\mathrm{e}^{-x}$ for $x\geq0$. Find the exact volume of the solid formed when the region between $D$ and the $x$-axis is rotated about the $x$-axis. You may use the fact that $x\mathrm{e}^{-2x}\to0$ as $x\to\infty$.

   [6 marks]

   Total for question 8: 12 marks

End of questions