Core Pure: Further calculusEdexcel A-Level Further Maths: Topic test

1. 1
   The curve $C$ has equation $y=e^{x/2}$ for $0\le x\le2$. The region $R$ is bounded by $C$, the coordinate axes and the line $x=2$.

   (a)

   Which integral gives the volume of the solid formed when $R$ is rotated through $2\pi$ radians about the $x$-axis?

   [1 mark]

   • A$\pi\int_0^2 e^{x/2}\,dx$
   • B$\pi\int_0^2 e^{x/4}\,dx$
   • C$\pi\int_0^2 e^{x}\,dx$
   • D$\pi\int_0^2 e^{2x}\,dx$

   (b)

   What is the exact volume of the solid formed in part (a)?

   [1 mark]

   • A$\pi\left(e^{2}-1\right)$
   • B$2\pi(e-1)$
   • C$\frac{\pi}{2}\left(e^{4}-1\right)$
   • D$\pi e^{2}$

   (c)

   The region $S$ is bounded by $C$, the $y$-axis and the line $y=e$. Show that the volume of the solid formed when $S$ is rotated through $2\pi$ radians about the $y$-axis is $4\pi\int_1^{e}(\ln y)^2\,dy$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $I=\int_0^{\infty}\frac{1}{x^2+9}\,dx$.

   (a)

   Which expression is equal to $\int_0^{R}\frac{1}{x^2+9}\,dx$, where $R$ is a positive constant?

   [1 mark]

   • A$3\arctan\frac{R}{3}$
   • B$\frac13\arctan\frac{R}{3}$
   • C$\frac13\arctan(3R)$
   • D$\frac13\ln\left(R^2+9\right)$

   (b)

   What is the value of $I$?

   [1 mark]

   • A$\frac{\pi}{2}$
   • B$\frac{\pi}{18}$
   • C$I$ does not exist because the range of integration is infinite
   • D$\frac{\pi}{6}$

   (c)

   Find the mean value of $f(x)=\frac{1}{x^2+9}$ over the interval $0\le x\le3$, giving your answer in terms of $\pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=\frac{3x^2-2x+10}{(x-2)\left(x^2+5\right)}$ for $x>2$.

   (a)

   Express $f(x)$ in the form $\frac{A}{x-2}+\frac{Bx+C}{x^2+5}$, where $A$, $B$ and $C$ are constants to be found.

   [3 marks]

   (b)

   Hence find the exact value of $\int_3^4 f(x)\,dx$, giving your answer in the form $\ln\left(k\sqrt6\right)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $y=x\arctan x-\frac12\ln\left(1+x^2\right)$ for $x\in\mathbb{R}$, and let $I=\int_0^1\frac{x^2}{\sqrt{4-x^2}}\,dx$.

   (a)

   Show that $\frac{dy}{dx}=\arctan x$, and hence find the exact value of $\int_0^1\arctan x\,dx$.

   [6 marks]

   (b)

   Use the substitution $x=2\sin\theta$ to find the exact value of $I$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A curve $C$ has parametric equations $x=t^2$, $y=2t$ for $0\le t\le2$. The region $R$ is bounded by $C$, the $x$-axis and the line $x=4$.

   (a)

   Which integral gives the volume of the solid formed when $R$ is rotated through $2\pi$ radians about the $x$-axis?

   [1 mark]

   • A$\pi\int_0^4 4t^2\cdot2t\,dt$
   • B$\pi\int_0^2 4t^2\,dt$
   • C$\pi\int_0^2 t^4\cdot2t\,dt$
   • D$\pi\int_0^2 4t^2\cdot2t\,dt$

   (b)

   What is the volume of the solid formed in part (a)?

   [1 mark]

   • A$16\pi$
   • B$32\pi$
   • C$64\pi$
   • D$128\pi$

   (c)

   The part of $R$ for which $0\le x\le k$ is rotated through $2\pi$ radians about the $x$-axis. Given that the volume of the solid formed is $18\pi$, find the value of $k$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $J=\int_0^8 x^{-2/3}\,dx$ and $K=\int_1^{\infty} x^{-3/2}\,dx$.

   (a)

   What is the value of $J$, given that the integrand is undefined at $x=0$?

   [1 mark]

   • A$6$
   • B$2$
   • C$12$
   • D$J$ does not exist

   (b)

   What is the value of $K$?

   [1 mark]

   • A$1$
   • B$K$ diverges because the upper limit is infinite
   • C$2$
   • D$0$

   (c)

   Find the mean value of $x^{-2/3}$ over the interval $1\le x\le8$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $h(x)=\frac{1}{x\left(x^2+1\right)}$ for $x>0$.

   (a)

   Express $h(x)$ in the form $\frac{A}{x}+\frac{Bx+C}{x^2+1}$, where $A$, $B$ and $C$ are constants to be found.

   [3 marks]

   (b)

   Hence show that $\int_1^{\infty}h(x)\,dx=\frac12\ln2$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=\frac{1}{\sqrt{16-x^2}}$ for $0\le x<4$.

   (a)

   The region $R$ is bounded by $C$, the coordinate axes and the line $x=2$. Find the exact volume of the solid formed when $R$ is rotated through $2\pi$ radians about the $x$-axis.

   [6 marks]

   (b)

   The integral $\int_0^4\frac{1}{\sqrt{16-x^2}}\,dx$ is improper. Explain why, evaluate the integral, and hence find the mean value of $y$ over the interval $0\le x\le4$.

   [6 marks]

   Total for question 8: 12 marks

End of questions