5.12 Areas and volumes of revolutionIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The curve $C$ has equation $y=x^2-4x$.

   (a)

   Find $\int_0^4(x^2-4x)\,dx$.

   [1 mark]

   • A$\frac{32}{3}$
   • B$-\frac{32}{3}$
   • C$\frac{64}{3}$
   • D$\frac{16}{3}$

   (b)

   Find the area of the region enclosed by $C$ and the $x$-axis for $0\le x\le4$.

   [1 mark]

   • A$-\frac{32}{3}$
   • B$\frac{64}{3}$
   • C$\frac{32}{3}$
   • D$\frac{16}{3}$

   (c)

   Use your GDC to find the total area enclosed between $C$ and the $x$-axis for $0\le x\le6$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For $k>0$, the region $R_k$ is enclosed by the curve $y=\sqrt{x}$, the $x$-axis and the line $x=k$. The region is rotated through $360^\circ$ about the $x$-axis to form a solid of volume $V$.

   (a)

   Which expression gives $V$?

   [1 mark]

   • A$\pi\int_0^k x\,dx$
   • B$\pi\int_0^k\sqrt{x}\,dx$
   • C$\pi\int_0^k x^2\,dx$
   • D$\pi\int_0^{\sqrt{k}} x\,dx$

   (b)

   Find the exact value of $V$ when $k=4$.

   [1 mark]

   • A$8$
   • B$\frac{16\pi}{3}$
   • C$16\pi$
   • D$8\pi$

   (c)

   Find the value of $k$ for which $V=18\pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $y=x^2$ is drawn for $x\ge0$. Distances are in centimetres. The region $S$ is enclosed by the curve, the $y$-axis and the line $y=9$.

   (a)

   The region $S$ is rotated through $360^\circ$ about the $y$-axis to model a bowl. Find the exact volume of the bowl.

   [3 marks]

   (b)

   A second solid is formed by rotating the region enclosed by the curve, the $x$-axis and the line $x=3$ through $360^\circ$ about the $x$-axis. Find its volume.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $K$ has equation $y=x^3-6x^2+8x$ for $0\le x\le4$. A GDC may be used.

   (a)

   (i) Show that $K$ crosses the $x$-axis at $x=0$, $x=2$ and $x=4$.
   (ii) Find $\int_0^4(x^3-6x^2+8x)\,dx$.
   (iii) Find the total area enclosed between $K$ and the $x$-axis.

   [6 marks]

   (b)

   The region enclosed by $K$ and the $x$-axis for $0\le x\le2$ is rotated through $360^\circ$ about the $x$-axis to form a solid.
   (i) Write down an integral for the volume $V$ of the solid.
   (ii) Use your GDC to find $V$.
   (iii) The region enclosed by $K$ and the $x$-axis for $2\le x\le4$ is also rotated about the $x$-axis. Explain, using the fact that $y(4-x)=-y(x)$, why this solid has the same volume as the first, and hence find the total volume of the two solids.

   [6 marks]

   Total for question 4: 12 marks

End of questions