Mathematics Extension 2 – Volumes

Volumes

Volume of Solids by Slicing

So far in the Mathematics and Mathematics Extension 1 course, we have done volume of solids of revolution by rotating a curve about the $Latex formula$ or $Latex formula$ axes. We now extend this idea to include rotations about lines other than the coordinate axes. Students are required to derive and find the volume of solids from first principles.

When finding the volume of solids of revolution by the method of slicing, students need to understanding that the total volume of the solid is formed by summing up infinitely thin slices that are perpendicular to the axis of rotation.

Students must show the following steps in order to get full marks

• Draw an accurate diagram
• Find an expression for the area of the cross-section of one slice
• Find an expression for the volume of a slice
• Find an expression for the total volume by summing up all the slices whilst taking the limit as the width approaches 0
• Solve the integral

The next example demonstrates what we already know from the Mathematics course.

Example 1

The region bounded by the curve $Latex formula$, the $Latex formula$-axis, and the lines $Latex formula$ and $Latex formula$ is rotated about the $Latex formula$-axis. Find the volume of the solid of revolution formed.

Solution 1

Area of cross section = $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

The following examples demonstrate how we can use the method of slicing to find the volume of the solid of revolution formed when a curve rotates about lines other than the coordinate axes.

Example 2

The region bounded by the curve $Latex formula$ and the $Latex formula$-axis is rotated about the line $Latex formula$. Find the volume of the solid of revolution.

Solution 2

Area of cross section = $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Area of cross section = $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Note that upper bound is $Latex formula$, because when $Latex formula$, $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 3

The region bounded by the curve $Latex formula$ and the $Latex formula$ and $Latex formula$ axes is rotated about the line $Latex formula$. Find the volume of the solid of revolution.

Solution 3

Area of cross section = $Latex formula$

$Latex formula$

$Latex formula$

Choose $Latex formula$ since we are using coordinate on the left branch

Area of cross section = $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Note that upper bound is $Latex formula$, because when $Latex formula$,$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 4

[HSC 2010, Question 3]

The region bounded by the $Latex formula$-axis and the curve $Latex formula$ is rotated about the line $Latex formula$. Find the volume generated.

Solution 4

Area of cross section = $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$ and $Latex formula$
Area of cross section = $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Note that upper bound is $Latex formula$, because when $Latex formula$, $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 5

[HSC 2008, Question 5]

Let $Latex formula$ and $Latex formula$ be constants, with $Latex formula$. A torus is formed by rotating the circle $Latex formula$ about the $Latex formula$-axis.

The cross-section at $Latex formula$, where $Latex formula$, is an annulus. The annulus has inner radius $Latex formula$ and outer radius $Latex formula$ where $Latex formula$ and $Latex formula$ are the roots of

$Latex formula$

(i) Find $Latex formula$ and $Latex formula$ in terms of $Latex formula$.

(ii) Find the area of the cross-section at height $Latex formula$, in terms of $Latex formula$.

(iii) Find the volume of the torus.

Solution 5

i)

$Latex formula$

Note that $Latex formula$

$Latex formula$

ii)

Area of cross section = $Latex formula$

$Latex formula$

$Latex formula$

iii)

$Latex formula$

$Latex formula$

$Latex formula$

Note that the definite integral represents the area in a semicircle with radius b

$Latex formula$

Example 6

(i) $Latex formula$ lies on the hyperbola $Latex formula$. Q is the foot of the perpendicular from P to the line $Latex formula$. If $Latex formula$ and $Latex formula$, show that $Latex formula$.

(ii) The region bounded by the hyperbola $Latex formula$ and the line $Latex formula$ is rotated through $Latex formula$ about the line $Latex formula$. Find the volume of the solid of revolution by slicing perpendicular to the axis of rotation.

Solution 6

i)

Using the formula for distance from a line:

$Latex formula$

$Latex formula$

$Latex formula$

But $Latex formula$

$Latex formula$

$Latex formula$

ii)

Area of cross-section $Latex formula$

$Latex formula$

Note that $Latex formula$ varies from $Latex formula$ to $Latex formula$ (this correspond the to the intersection points of $Latex formula$ and $Latex formula$, and $Latex formula$ and $Latex formula$ and their distance from the origin)

$Latex formula$

$Latex formula$

$Latex formula$

=$Latex formula$

$Latex formula$

Volume of Solids by Cylindrical Shells

Another method of finding the volume of a solid of revolution is by summing up cylindrical shells that are parallel to the axis of rotation. The reason behind multiple methods is because often one method gives elegant solutions to problems which would be challenging to solve by another.

There are two methods of using cylindrical shells:

1. The first principle method where the volume of the solid of revolution is found by summing up thin cylindrical shells.
2. Treating the cylindrical shell as a rectangular prism by rolling out the cylinder.

Both methods are acceptable by the syllabus but the second method tends to be easier, thus it will be used more often.

The following examples demonstrate how we can use the method of cylindrical shell to find the volume of the solid of revolution.

Example 7

[HSC 2007, Question 3]

Use the method of cylindrical shells to find the volume of the solid formed when the shaded region bounded by

$Latex formula$

is rotated about the $Latex formula$-axis.

Solution 7

Consider a thin cylindrical shell parallel to axis of rotation. Since the shell has very small width, its volume can be approximated as a rectangular prism. For any arbitrary point (x,y) on the curve, the resulting prism will have:

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 8

[HSC 2009, Question 3]

The diagram shows the region enclosed by the curves $Latex formula$ and $Latex formula$.

The region is rotated about the-axis.

Use the method of cylindrical shells to find the volume of the solid formed.

Solution 8

(We will use first principles method here)

Consider a very thin cylindrical shell,

$Latex formula$

where $Latex formula$ is the $Latex formula$ coordinate of the parabola, and $Latex formula$ is the coordinate for the straight line.

$Latex formula$

$Latex formula$

Because $Latex formula$ is so small, and taking its square makes it even smaller, we approximate $Latex formula$.

$Latex formula$$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 9

[HSC 2006, Question 5]

A solid is formed by rotating the region bounded by the curve $Latex formula$ and the line $Latex formula$ about the $Latex formula$-axis. Use the method of cylindrical shells to find the volume of this solid.

Solution 9

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 10

[HSC 2005, Question 4]

The shaded region between the curve $Latex formula$, the $Latex formula$-axis, $Latex formula$ and $Latex formula$ the lines and, where $Latex formula$, is rotated about the $Latex formula$-axis to form a solid of revolution.

(i) Use the method of cylindrical shells to find the volume of this solid in terms of $Latex formula$.

(ii) What is the limiting value of this volume as $Latex formula$?

Solution 10

i)

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

ii)

$Latex formula$

Example 11

[HSC 2011, Question 7]

The diagram shows the graph of $Latex formula$ for $Latex formula$.

The area bounded by $Latex formula$, the line $Latex formula$ and the $Latex formula$-axis is rotated about the line $Latex formula$ to form a solid.

Use the method of cylindrical shells to find the volume of the solid.

Solution 11

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 12

The region bounded by the curve $Latex formula$ and the $Latex formula$ and $Latex formula$-axes is rotated about the line $Latex formula$. Find the volume of the solid of revolution.

Solution 12

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Volume of Solids with Parallel Cross Sections of Similar Shapes

So far we have studied volumes by rotating a 2-dimensional region about a line. Now we will look at volumes of solids of irregular shape but with similar cross sections.

Students must show the following steps in order to get full marks

• Draw an accurate diagram
• Find an expression for the area of an arbitrary cross-section
• Find an expression for the volume of the cross-section
• Find an expression for the total volume by summing up all the slices of the cross-section whilst taking the limit as the width approaches 0
• Solve the integral

The following examples demonstrate this concept.

Example 13

[HSC 2006, Question 4]

The base of a solid is the parabolic region $Latex formula$ shaded in the diagram.

Vertical cross-sections of the solid perpendicular to the $Latex formula$-axis are squares.

Find the volume of the solid.

Solution 13

Area of cross-section = $Latex formula$

Volume of cross-section: $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 14

[HSC 2011, Question 3]

The base of a solid is formed by the area bounded by $Latex formula$ and $Latex formula$ for $Latex formula$.

Vertical cross-sections of the solid taken parallel to the $Latex formula$-axis are in the shape of isosceles triangles with the equal sides of length 1 unit as shown in the diagram.

Find the volume of the solid.

Solution 14

Area of cross-section = $Latex formula$

Volume of cross-section: $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 15

[HSC 2007, Question 4]

The base of a solid is the region bounded by the curve $Latex formula$, the $Latex formula$-axis and the lines $Latex formula$ and $Latex formula$, as shown in the diagram.

Vertical cross-sections taken through this solid in a direction parallel to the $Latex formula$-axis are squares. A typical cross-section, PQRS, is shown.

Find the volume of the solid.

Solution 15

Area of cross-section = $Latex formula$

Volume of cross-section: $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 16

[HSC 2010, Question 6]

The diagram shows the frustum of a right square pyramid. (A frustum of a pyramid is a pyramid with its top cut off.)

The height of the frustum is $Latex formula$ m. Its base is a square of side $Latex formula$ m, and its top is a square of side $Latex formula$ m (with$Latex formula$).

A horizontal cross-section of the frustum, taken at height $Latex formula$ m, is a square of side $Latex formula$ m, shown shaded in the diagram.

(i) Show that $Latex formula$.

(ii) Find the volume of the frustum.

Solution 16

i)

Consider the triangle formed from the edges, as shown below:

Noting that they are similar triangles, we have:

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

ii)

Area of cross-section = $Latex formula$

Volume of cross-section: $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 17

[HSC 2009, Question 6]

The base of a solid is the region enclosed by the parabola $Latex formula$ and the $Latex formula$-axis. The top of the solid is formed by a plane inclined at $Latex formula$ to the $Latex formula$-plane. Each vertical cross-section of the solid parallel to the $Latex formula$-axis is a rectangle. A typical cross-section is shown shaded in the diagram.

Find the volume of the solid.

Solution 17

Area of cross-section = $Latex formula$

Volume of cross-section: $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

Example 18

[HSC 2005, Question 5]

The base of a right cylinder is the circle in the $Latex formula$-plane with centre $Latex formula$ and radius 3. A wedge is obtained by cutting this cylinder with the plane through the $Latex formula$-axis inclined at $Latex formula$ to the $Latex formula$-plane, as shown in the diagram.

A rectangle slice ABCD is taken perpendicular to the base of the wedge at a distance $Latex formula$ from the $Latex formula$-axis.

(i) Show that the area of ABCD is given by $Latex formula$.

(ii) Find the volume of the wedge.

Solution 18

i)

Area = $Latex formula$

ii)

Volume of cross-section: $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$