Volumes of Revolution (Edexcel A Level Further Maths: Core Pure): Exam Questions

Figure 1

A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student’s design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the $y$-axis, the $x$-axis, the line with equation $x=1$, the curve $C_1$ and the curve $C_2$ through 360° about the $y$-axis. The point A has coordinates (1, 0.5) and the point B has coordinates (0.5, 2.5) where the units are centimetres.

The curve $C_1$ is modelled by the equation

$x=\frac{a}{y+b}0.5\le y\le 2.5$

Determine the value of $a$ and the value of $b$ according to the model.

The curve $C_2$ is modelled to be an arc of the circle with centre (0, 3).

Use calculus to determine the volume of plastic required to make the chess piece according to the model.

Figure 2 shows the image of a gold pendant which has height 2cm. The pendant is modelled by a solid of revolution of a curve $C$ about the $y$-axis. The curve $C$ has parametric equations

$x=\cos \theta +\frac{1}{2}\sin 2\theta ,y=-(1+\sin \theta )0\le \theta \le 2\pi$

Show that a Cartesian equation of the curve $C$ is

$x^2=-(y^4+2y^3)$

Hence, using the model, find, in cm³, the volume of the pendant.

Figure 1 shows the vertical cross-section of a wooden table leg, which is modelled as a solid of revolution about the $x$-axis (the axis of the leg), where $O$ is the fixed origin. The leg is formed from a plain cylindrical section together with a curved decorative section.

The circular cross-section at each end of the curved section has diameter 8 cm. The curved section has length 5 cm, and the cylindrical section, which has the same radius as the ends of the curved section, has length 3 cm.

The curved section is modelled by the curve $CD$ with parametric equations

$y=a+2\text{sin}2t,x=b\text{cos}t,0\le t\le \frac{\pi }{2}$

where $a$ and $b$ are constants.

Determine the value of $a$ and the value of $b$ according to the model.

Using algebraic integration and showing all your working, determine the volume of the table leg, giving your answer to the nearest cm³.

State a limitation of the model.