FM Volumes of revolution II name _______________________

Objective Deadlines / Progress

Volu

mes

aga

inst

x-a

xis a

nd y

-axi

s

Apply the formulas for volumes of revolution against the x-axis and y-axis

Apply volumes of revolution to complex functions including trig functions, exponential and natural logarithm functions Know some standard volumes of revolution such as volume of a sphere or cone

Apply integration methods such as substitution and integration by parts

Volu

mes

with

par

amet

ric

equa

tions

Find areas under a curve given as parametric functions

Find volumes under a curve given as parametric functions against both xAxis and yAxis

Mod

ellin

g

Solve problems in context including using scaled models

FM Volumes of revolution II name _______________________

Volume of revolution around x-axis

Notes

The volume of revolution formed when y=f(x) is rotated around the x-axis between the x-axis, x=a and x=b is given by

Volume=π∫a

b

y2dx

Prior knowledge: You should to be able to find integrals using substitution methods; trigonome4tric identities and integration by parts

WB A1 The region R is bounded by the curve y=ex, the x-axis and the vertical lines x=0 and x=2Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give an exact answer

WB A2 The region R is bounded by the curve y=e2 x, the x-axis and the vertical lines x = 0 and x = 4. Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Write your answer as a multiple of π

FM Volumes of revolution II name _______________________

WB A3 The region R is bounded by the curve y=sec x, the x-axis and the vertical lines x=π6 and

x=π3

Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give an exact answer

WB A4 The region R is bounded by the curve y=sin2 x, the x-axis and the vertical lines x = 0 and x = π/2 Find the volume of the solid formed when the region is rotated 2π radians about the x-axis. Give your answer as a multiple of π2

FM Volumes of revolution II name _______________________

Notes

The volume of revolution formed when x=f(y) is rotated around the y-axis between the y-axis, y=a and y=b is given by

Volume=π∫a

b

x2dx

When you use this formula you are integrating with respect to y. So you may need to rearrange functions accordingly

WB B1 The region R is bounded by the curve y=4 ln x−1, the y-axis, x-axis and the horizontal lines y = 0 and y = 4 Show that the volume of the solid formed when the region is rotated 2π radians about the y-axis is 2π √e (e2−1 )

FM Volumes of revolution II name _______________________

FM Volumes of revolution II name _______________________

WB B2 The region R is bounded by the curve y=x2−2the y-axis and the vertical lines y=1 and y=3Find the volume of the solid formed when the region is rotated 2π radians about the y-axis. Give your answer as a multiple of π

WB B3 The area bounded by the curve y=x2 and the lines x=3 and y=1 is rotated 2π about the line y=1 Find the volume of the solid formed

Can you generalise to give a formula for the volume formed when the curve is rotated about line y=a

FM Volumes of revolution II name _______________________

Volumes of revolution and Parametric Functions

Notes

When a curve is given in parametric equations we can use the following

Area=∫ y dx=∫ y dxdtdt

The volume of revolution formed when the parametric curve is rotated around the x-axis is given by

Volume=π ∫x=a

x=b

y2dx=π ∫t=q

t= p

y (t)2 dxdtdt

The corresponding volume of revolution formed when the parametric curve is rotated around the y-axis is given by

Volume=π ∫y=a

y=b

x2dy=π ∫t=q

t= p

x (t )2 dydtdt

Make sure you find the new bounds, p and q, of the integral

Areas with parametric functions

WB C1 Find the area under the curve given by the parametric equations:

x=2 t+1 y=t3−1t1≤t ≤2

WB C2 Find the area under the curve given by the parametric equations:

FM Volumes of revolution II name _______________________

x=sin θ y=cosθ 0≤θ≤ π2

Volumes with parametric functions

WB C3 The curve C has parametric equations:

x=t (1+t ), y= 11+t , t ≥0

The region R is bounded by C, the x-axis and the lines x = 0 and x = 2. Find the volume of the solid formed when R is rotated 2π radians about the x-axis.

FM Volumes of revolution II name _______________________

WB C4 Find the volume of revolution formed by rotating the curve x=cos t, y=√sin t−1 ,

0≤ t ≤ π2 by 2 around the x-axis

WB C5 Find the volume of revolution formed by rotating the curve x=e t, y=√t−1 , 2≤t ≤2π about the x axis

FM Volumes of revolution II name _______________________

WB C6 Find the exact volume of revolution formed by rotating the curve x=√ t−2, y=t2 , 2≤t ≤3 about the yAxis

WB C7 Find the exact volume of revolution formed by rotating the curve x=sin t , y=2t ,

FM Volumes of revolution II name _______________________

0≤ t ≤ π3 About the yAxis. Give your answer in terms of π

FM Volumes of revolution II name _______________________

Volumes of revolution and modelling

WB D1 The diagram shows a model of a goldfish bowl. The cross-section of the model is described by the curve with parametric equations

x=2sin t , and y=2cos t π6≤ t ≤ 11 π

6Where the units of x and y are given in cm. The goldfish bowl volume is formed by rotating the curve around the y-axis to form a solid of revolution.a) Find the volume of the water required to fill the model to a height of 3 cmb) The real goldfish bowl has a maximum diameter of 48 cm. Find the volume of water needed to fill the real bowl to a corresponding height.

4 cm

3 cm

FM Volumes of revolution II name _______________________

FM Volumes of revolution II name _______________________

WB D2 The diagram shows the image of a gold pendant which has height 2 cm. the pendant is modelled by a solid of revolution of a curve C about the y-axis. The curve

has parametric equations x=cos t+ 12sin 2 t and y=−(1+sin t ) ,0≤t ≤2π

a) Show that a Cartesian equation of the curve C is x2=−( y 4+2 y3 ) (4)b) Hence, using the model, find in cm3, the volume of the pendant (4)