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Aims:• To know what a volume of revolution is and learn where the formula comes from.• To be able to calculate a volume of revolution about the x-axis• To practice sketching curves, using the calculator if you wish. • To be able to calculate a volume of revolution about the y-axis• Practice selecting the most appropriate method of integration

Volumes of revolution Lesson 1

Starter choose method to integrate - ppt

Volumes of revolution

Consider the area bounded by the curve y = f(x), the x-axis and x = a and x = b.

If this area is rotated 360° about the x-axis a three-dimensional shape called a solid of revolution is formed.

The volume of this solid is called its volume of r_____________.

Volumes of revolutionWe can calculate the volume of revolution by dividing the volume of revolution into thin slices of width δx.

The volume of each slice is approximately cylindrical, of radius y and height δx, and is therefore approximately equal to

=2

=

x b

x a

V y x As δx x b

x x a

V y x

=

2

0 =

= lim

Volumes of revolution

In general, the volume of revolution V of the solid generated by rotating the curve y = f(x) between x = a and x = b about the x-axis is:

Volumes of revolution are usually given as multiples of π.

2=b

aV y dx

Volumes of revolution

Find the volume of the solid formed by rotating the area between the curve y = x(2 – x), the x-axis, x = 0, and x = 2 360° about the x-axis.

2 2

0=V y dx

2 2 2

0= (2 )x x dx

1. Exercise A page 126Qu 1, 2, 3a, c, e, g

On w/b

Find the volume of the solid formed by rotating the area between the curve y = 2/x, the x-axis, x = 2, and x = 6 360° about the x-axis.

1. Exercise A page 126Qu 1, 2, 3a, c, e, g

Volumes of revolution

Similarly, the volume of revolution V of the solid generated by rotating the curve x = f(y) between y = a and y = b about the y-axis is:

2=b

aV x dy

Volumes of revolution

y

yV x dy

2

1

2=

Find the volume of the solid formed by rotating the area between the curve y = , the y-axis, x = 1, and x = 1/2 360° about the y-axis.

1x

As Rearranging y = gives x = .1

x1y

x1 2 3 4

y

-1

1

2

3

On w/b

Find the volume of the solid formed by rotating the area between the curve y = x3 - 1, the y-axis, x = 2, and x = 4 2 radians about the y-axis.

2. Match up cards in groups 3 or 43. Exercise B page 128 qu 2, 4, 64. Then mixed exercise page 128 qu 1 - 5